I don't see a question in your message. Beside, your code needs a library to run, so I moved it.

" I keep getting various errors (most frequently "null pointer exception")"
This is unhelpful. We don't have the full code (at least the imports), nor the spreadsheet, so we cannot run the code. You cannot expect us to "guess" ¹ what are these "various" errors, or where the NPEs happen.
Give more information, please.

¹Sometime I can "guess", seeing some blatant errors, but here the code seems globally OK from a quick look.

Okay, sorry about that. I didn't post the full code because it's pulling data from a private spreadsheet, and wasn't sure if you'd be able to run it anyways. Here it is though, in addition to the full error code (this is mostly taken from a tutorial on blog.blprnt.com). If it helps, the spreadsheet has two columns: one for age (obviously a number), and sex (M or F). Occasionally, there are blank cells where we don't have all the information. I highlighted the line that throws the error; look at the bottom for the full text of the error.

1. //This is the Google spreadsheet manager and the id of the spreadsheet that we want to populate, along with our Google username & password
   SimpleSpreadsheetManager sm;
   String sUrl = "pEAyAv6KI8MBIFtOls_ZrMA";
   String googleUser = "guser";
   String googlePass = "gpwd";
    
   void setup() {
     //This code happens once, right when our sketch is launched
    size(800,600);
    background(0);
    smooth();
   
    int[] numbers = getNumbers();
    String[] sex = getSex();
    
    fill(255,40);
    noStroke();
    for (int i = 0; i < numbers.length; i++) {
   
      if( (sex[i].equals("F")) && (numbers[i] > 0) ) {  //This is the line that gets highlighted when the error triggers
        ellipse(numbers[i] * 8, width/2, 8,8);
      };
   
    };
    
   };
   
   void draw() {
     //This code happens once every frame.
   };
2.  //Function to get an Array of integers from a Google Spreadsheet
     int[] getNumbers() {
         println("Asking Google for numbers...");
         sm = new SimpleSpreadsheetManager();
         sm.init("RandomNumbers", googleUser, googlePass);
         sm.fetchSheetByKey(sUrl, 0);
        
         int n = sm.currentListEntries.size();
         int[] returnAge = new int[n];
        
         for (int i = 0; i < n; i++) {
           if( sm.getCellValue(5,i) != null ) {
             returnAge[i] = int(sm.getCellValue(5, i));
           };
         };
         return(returnAge);
     };
    
     String[] getSex() {
         println("Asking Google for sex...");
         sm = new SimpleSpreadsheetManager();
         sm.init("Sex", googleUser, googlePass);
         sm.fetchSheetByKey(sUrl, 0);
        
         int m = sm.currentListEntries.size();
         String[] returnSex = new String[m];
        
         for (int i = 0; i < m; i++) {
           if( sm.getCellValue(6,i) != null ) {
             returnSex[i] = sm.getCellValue(6, i);
           };
         };
   
         return(returnSex);
     };
3. /*
   
   SimpleSpreadsheetManager Class
   blprnt@blprnt.com
   July, 2009
   
   This is a quick & dirty class for accessing data from the Google Spreadsheet API.
   
   
   */
   
   import com.google.gdata.client.spreadsheet.*;
   import com.google.gdata.data.*;
   import com.google.gdata.data.spreadsheet.*;
   import com.google.gdata.util.*;
   
   import java.io.BufferedReader;
   import java.io.IOException;
   import java.io.InputStreamReader;
   import java.io.PrintStream;
   import java.net.URL;
   import java.util.HashMap;
   import java.util.List;
   import java.util.Map;
   
   
   public class SimpleSpreadsheetManager {
   
     SpreadsheetService myService;
     SpreadsheetEntry spreadsheetEntry;
     SpreadsheetFeed sheetFeed;
     WorksheetEntry worksheetEntry;
   
     List spreadsheets;
   
     String user;
     String pass;
    
     ListFeed currentListFeed;
     CellFeed currentCellFeed;
     List currentCells;
     List currentListEntries;
    
     int currentTotalRows;
     int currentTotalCols;
    
     String currentTitle;
     String[] tagArray;
    
     URL listFeedUrl;
   
   
     SimpleSpreadsheetManager() {
      
     };
    
     /*
    
     INIT FUNCTION
     Opens session, uses username & password for authentication
    
     */
   
     void init(String sessionName, String u, String p) {
       user = u;
       pass = p;
       myService = new SpreadsheetService(sessionName);
       try {
         myService.setUserCredentials(user, pass);
       }
       catch (Exception e) {
         println("ERROR IN AUTHENTICATION");
       };
   
     };
    
     /*
    
     FETCH SHEET BY KEY
     Gets a spreadsheet listfeed using the unique key - this is in the URL of the spreadsheet
     The retrieved sheet is both returned and set as the currentListFeed
    
     */
    
     ListFeed fetchSheetByKey(String k, int wi) {
      
       ListFeed f = new ListFeed();
       CellFeed cf = new CellFeed();
       WorksheetFeed w = new WorksheetFeed();
      
       //GET WORKSHEETS FEED
       try {
         URL worksheetFeedUrl = new URL("http://spreadsheets.google.com/feeds/worksheets/" + k + "/private/full");
         WorksheetFeed wf2 = new WorksheetFeed();
         w = myService.getFeed(worksheetFeedUrl, wf2.getClass());
       }
       catch(Exception e) {
         println("ERROR RETRIEVING WORKSHEET FEED");
       };
      
       List worksheets = w.getEntries();
       WorksheetEntry we = (WorksheetEntry) worksheets.get(wi);  
       println("RETRIEVED WORKSHEET " + we.getTitle().getPlainText());
      
       //GET LIST FEED URL
       try {
         listFeedUrl = we.getListFeedUrl();//new URL("http://spreadsheets.google.com/feeds/list/" + k + "/od6/private/full");
         ListFeed lf2 = new ListFeed();
         f = myService.getFeed(listFeedUrl, lf2.getClass());
       }
       catch(Exception e) {
         println("ERROR RETRIEVING LIST FEED");
       };
      
       //GET CELL FEED
       try {
         URL cellFeedUrl = we.getCellFeedUrl();//new URL("http://spreadsheets.google.com/feeds/cells/" + k + "/od6/private/full");
         CellFeed lf2 = new CellFeed();
         cf = myService.getFeed(cellFeedUrl, lf2.getClass());
   
       }
       catch(Exception e) {
         println("ERROR RETRIEVING LIST FEED");
       };
      
       currentListFeed = f;
       currentCellFeed = cf;
       currentCells = cf.getEntries();
       currentListEntries = f.getEntries();
      
       currentTitle = we.getTitle().getPlainText();
       currentTotalRows = currentListEntries.size();
      if (currentListEntries.size() > 0) {
       ListEntry le = (ListEntry) currentListEntries.get(0); 
       currentTotalCols = le.getCustomElements().getTags().size();
   
       Set<String> tags = le.getCustomElements().getTags();
       tagArray = new String[tags.size()];
       tagArray = tags.toArray(tagArray);
      };
      
       return(f);
     };
    
     /*
    
     GET CELL VALUE
     Returns the value held in an individual sheet cell.
    
     */
    
     String getCellValue(int c, int r) {
      
       ListEntry le = (ListEntry) currentListEntries.get(r);   
       Set<String> tags = le.getCustomElements().getTags();
       String[] tagArray = new String[tags.size()];
       tagArray = tags.toArray(tagArray);
      
       return(le.getCustomElements().getValue(tagArray[c]));
     
     };
    
     String getCellValue(String tag, int r) {
    
       ListEntry le = (ListEntry) currentListEntries.get(r);   
       return(le.getCustomElements().getValue(tag));
     
     };
    
     void setCellValue(String tag, int r, String val) {
    
       ListEntry le = (ListEntry) currentListEntries.get(r);   
       le.getCustomElements().setValueLocal(tag, val);
       try {
         ListEntry updatedRow = le.update();
       }
       catch (Exception e){
        
       };
     
     };
    
     void setNewCellValue(String tag, int r, String val) {
    
       ListEntry le = new ListEntry(); 
      
       try {
        le.getCustomElements().setValueLocal(tag, val);
        ListEntry insertedRow = myService.insert(listFeedUrl, le);
         ListEntry updatedRow = insertedRow.update();
       }
       catch (Exception e){
        
       };
     
     };
    
   };

This is the error code that's printed:

1. Asking Google for numbers...
   RETRIEVED WORKSHEET Current 2.0
   Asking Google for sex...
   RETRIEVED WORKSHEET Current 2.0
   processing.app.debug.RunnerException: NullPointerException
       at processing.app.Sketch.placeException(Sketch.java:1543)
       at processing.app.debug.Runner.findException(Runner.java:582)
       at processing.app.debug.Runner.reportException(Runner.java:558)
       at processing.app.debug.Runner.exception(Runner.java:498)
       at processing.app.debug.EventThread.exceptionEvent(EventThread.java:367)
       at processing.app.debug.EventThread.handleEvent(EventThread.java:255)
       at processing.app.debug.EventThread.run(EventThread.java:89)
   Exception in thread "Animation Thread" java.lang.NullPointerException
       at FT_age.setup(FT_age.java:100)
       at processing.core.PApplet.handleDraw(Unknown Source)
       at processing.core.PApplet.run(Unknown Source)
       at java.lang.Thread.run(Thread.java:619)
   

Thanks a lot for the help!

On this line, it means that sex[i] isn't initialized. You should println() the steps (indexes, what you get, etc.) to analyze the progress of the algorithm and see where it fails.

Okay - not sure if I did it right, but I changed this block:

1. if( (sex[i].equals("F")) && (numbers[i] > 0) ) {
        ellipse(numbers[i] * 8, width/2, 8,8);
      };

To this:

1. println(sex[i] + ", " + numbers[i] + " years old.");
      /*
      if( (sex[i].equals("F")) && (numbers[i] > 0) ) {
        ellipse(numbers[i] * 8, width/2, 8,8);
      };
      */

All I did was comment out the offending if statement and print both variables. This runs just fine, and returns this input:

1. Asking Google for numbers...
   RETRIEVED WORKSHEET Current 2.0
   Asking Google for sex...
   RETRIEVED WORKSHEET Current 2.0
   null, 0 years old.
   F, 63 years old.
   M, 78 years old.
   M, 40 years old.
   F, 43 years old.
   F, 56 years old.
   F, 42 years old.
   M, 46 years old.
   F, 59 years old.
   M, 67 years old.
   M, 63 years old.
   M, 52 years old.
   null, 0 years old.
   F, 63 years old.
   F, 49 years old.
   F, 51 years old.
   F, 43 years old.
   F, 57 years old.
   F, 36 years old.
   F, 49 years old.
   M, 57 years old.
   F, 67 years old.
   F, 48 years old.
   F, 48 years old.
   F, 66 years old.
   F, 56 years old.
   M, 49 years old.
   M, 41 years old.
   M, 61 years old.
   F, 61 years old.
   F, 54 years old.
   F, 30 years old.
   F, 25 years old.
   F, 45 years old.
   F, 35 years old.
   F, 59 years old.
   M, 58 years old.
   F, 45 years old.
   M, 56 years old.
   F, 47 years old.
   F, 46 years old.
   F, 53 years old.
   F, 47 years old.
   M, 36 years old.
   F, 38 years old.
   F, 26 years old.
   F, 39 years old.
   F, 43 years old.
   F, 31 years old.
   M, 68 years old.
   F, 59 years old.
   F, 44 years old.
   F, 69 years old.
   F, 60 years old.
   M, 49 years old.
   F, 26 years old.
   F, 56 years old.
   F, 44 years old.
   F, 48 years old.
   F, 46 years old.
   M, 40 years old.
   F, 52 years old.
   F, 52 years old.
   F, 64 years old.
   M, 62 years old.
   M, 46 years old.
   F, 44 years old.
   M, 55 years old.
   F, 67 years old.
   M, 55 years old.
   F, 30 years old.
   F, 56 years old.
   M, 63 years old.
   F, 62 years old.
   M, 56 years old.
   M, 44 years old.
   F, 30 years old.
   M, 27 years old.
   M, 39 years old.
   F, 24 years old.
   M, 54 years old.
   M, 0 years old.
   F, 65 years old.
   F, 0 years old.
   F, 59 years old.
   F, 47 years old.
   M, 32 years old.
   M, 35 years old.
   M, 51 years old.
   F, 19 years old.
   F, 21 years old.
   M, 53 years old.
   M, 69 years old.
   M, 37 years old.
   F, 35 years old.
   F, 70 years old.
   F, 56 years old.
   M, 41 years old.
   M, 38 years old.
   F, 30 years old.
   F, 67 years old.
   F, 69 years old.
   F, 52 years old.
   F, 0 years old.
   F, 53 years old.
   F, 32 years old.
   F, 51 years old.
   M, 41 years old.
   F, 54 years old.
   M, 39 years old.
   F, 55 years old.
   F, 50 years old.
   F, 44 years old.
   M, 45 years old.
   M, 32 years old.
   F, 43 years old.
   M, 40 years old.
   F, 64 years old.
   F, 62 years old.
   F, 54 years old.
   M, 0 years old.
   F, 46 years old.
   F, 44 years old.
   F, 68 years old.
   M, 36 years old.
   M, 47 years old.
   M, 35 years old.
   M, 44 years old.
   M, 39 years old.
   M, 49 years old.
   F, 53 years old.
   F, 60 years old.
   F, 40 years old.
   F, 27 years old.
   F, 50 years old.
   F, 29 years old.
   F, 34 years old.
   F, 69 years old.
   F, 67 years old.
   F, 39 years old.
   M, 71 years old.
   F, 25 years old.
   F, 23 years old.
   F, 23 years old.
   M, 55 years old.
   F, 54 years old.
   F, 23 years old.
   M, 65 years old.
   F, 82 years old.
   M, 35 years old.
   F, 31 years old.
   F, 61 years old.
   F, 53 years old.
   F, 66 years old.
   F, 63 years old.
   M, 58 years old.
   M, 45 years old.
   M, 39 years old.
   M, 48 years old.
   F, 57 years old.
   F, 52 years old.
   M, 38 years old.
   M, 38 years old.
   F, 35 years old.
   F, 19 years old.
   M, 66 years old.
   F, 48 years old.
   F, 55 years old.
   F, 0 years old.
   F, 64 years old.
   F, 79 years old.
   M, 32 years old.
   F, 64 years old.
   F, 58 years old.
   M, 30 years old.
   F, 71 years old.
   M, 55 years old.
   F, 41 years old.
   M, 43 years old.
   M, 55 years old.
   F, 49 years old.
   F, 28 years old.
   M, 41 years old.
   F, 42 years old.
   F, 56 years old.
   F, 66 years old.
   F, 31 years old.
   F, 35 years old.
   null, 0 years old.
   null, 0 years old.
   null, 0 years old.
   null, 0 years old.
   null, 0 years old.
   null, 0 years old.
   null, 0 years old.
   null, 0 years old.
   null, 0 years old.
   null, 0 years old.
   null, 0 years old.
   null, 0 years old.
   null, 0 years old.
   null, 0 years old.
   

I don't understand why it prints these two arrays completely fine, but the if statement throws an error. I really appreciate your help!

The arrays are not "completely fine", the "null, 0 years old." entries are the culprit.
A classical way to handle the problem is to do:

1. if (sex[i] != null && sex[i].equals("F") && numbers[i] > 0) {
2.    ellipse(numbers[i] * 8, width/2, 8, 8);
3. }

Note: no need for semi-colons after a closing brace (in general).

Okay, thanks a bunch. That solved it!