Yes.
http://processing.org/reference/selectInput_.html
You can then open the file with loadStrings() or loadBytes() like you normally would.

So what I'm imagining is that in some programming languages it might be possible to have code called when, say, a jpeg file is double clicked in Finder. Perhaps something along the lines of "void setupAndOpen(String fileName)" or something like that?

That's different. If you want to setup your computer so that certain filetypes are opened by a processing application, that is a system preference that I'm sure you can set. For me (a Windows user), I pick "Open With..." then "Select program from list" and then I have to navigate to the sketch that I want to use to open up that filetype. Then there's a checkbox I can select to "Always use this program to open this type of file". or something.

Point is: It's a system setting. You can't set this setting via processing.

Ahhh. Ahhhh. Ok. Yes, that is a good question. You need your sketch to know the name of the file that your application was told to use. Right. My guess is that the name of the file would be passed as a command line argument to your sketch, in which case 
http://wiki.processing.org/w/Setting_width/height_dynamically
is relevant.

If it's not part of the command line arguments, then I'm at a loss. Phi.lo would know...

There is no magic, you are still dependent on the way you defined the .rtx association to your sketch.
For example, in Windows, you will define it as:
path/to/your/sketch/Sketch.exe %1
(for a sketch named... Sketch, of course!).
in the registry. The %1 is there to indicate to pass the argument (the name of the .rtx file) to the command line.

I was annoyed we had to manually put the name of the class in the code...
I found a trick to get this automatically.
Here is the revised code:

1. // Based on http://wiki.processing.org/w/Setting_width/height_dynamically
2.  
3. // Static to be accessed by main()
4. static String[] newArgs;
5.  
6. // Overload the PApplet's main
7. public static void main(String args[])
8. {
9.   newArgs = new String[args.length+1];
10.  
11.   // Trick found at http://stackoverflow.com/questions/41894/0-program-name-in-java-discover-main-class
12.   StackTraceElement[] stack = Thread.currentThread ().getStackTrace();
13.   StackTraceElement main = stack[stack.length - 1];
14.   String mainClass = main.getClassName();
15.  
16.   // PApplet wants the name of the class as first argument
17.   newArgs[0] = mainClass;
18.   // Copy the remainder of the parameters given on the command line
19.   arrayCopy(args, 0, newArgs, 1, args.length);
20.   // Call the original PApplet main with the build arguments
21.   PApplet.main(newArgs);
22. }
23.  
24. void setup()
25. {
26.   size(400, 400);
27.   background(255);
28.   fill(0);
29.   // Now we can access the global array
30.   println(newArgs.length);
31.   for (int i = 0; i < newArgs.length; i++)
32.   {
33.     text(newArgs[i], 10, 15 * i + 20);
34.   }
35. }

I don't know how file associations are managed in MacOS, but the general idea is that if you look at the way it is defined, it should show the path to your executable sketch, and add a parameter after it, meaning to pass the name (path) of the associated file to the application. On a Unix system, it should be $1 instead of %1%.
With regard with your problem with the Apple version of Java, you can try and add a line like:
println(stack);
I suppose Apple added a layer in the stack. The relevant line looks like:
CommandLineTest.main(CommandLineTest.java:31)
on my computer. (CommandLineTest is that part holding the sketch name). Perhaps you can access this name with StackTraceElement main = stack[stack.length - 2]; or similar (nth element from the bottom).

I fail to understand how your OpenFile class is related to this topic. What problem does it solves? It looks like File. list() which is standard Java (cross-platform).

I fixed the code! I'm not exactly sure how. But I updated the code in my previous post to the working one.