Some suggestions:

• Remove unused code/variables from your sketch.
  
• Use autoformat (CTRL + T), otherwise your code becomes hard to read.
  
• paused is also a boolean used in PApplet, so I would advise to use another name for your variable.
• Your class increases the yloc in three places (!), so you need to encapsulate all three within pause.
• Your class has switch1 and switch2 variables that (once toggled) keep moving shapes regardless.
  

Perhaps you can try to compartimentalize your code more, so it's behavior will be easier to understand, predict and alter. Currently it's hard to read and not really clear what's happening where. I suppose in your actual sketch this will be an even bigger issue.

Here are a few adaptations in your simplified code example, that implement a pause functionality.

Adapted Code

1. Head [] en = new Head[500];
2. Head [] po = new Head[500];
3. int t = 1000;
4. float speed = 1;
5. boolean pause;
6. 
   
7. void setup() {
8.   size(700, 600);
9.   for (int i=0; i<en.length; i++) { en[i] = new Head(); }
10.   for (int i=0; i<po.length; i++) { po[i] = new Head(); }
11. }
12. 
    
13. void draw() {
14.   background(0);
15.   redd();
16.   greenn();
17.   britva();
18. }
19. 
    
20. void redd() {
21.   int i = 0;
22.   for (t=0; millis() > t; t=t+1950) {
23.     en[i].appear(2);
24.     i++;
25.   }
26. }
27. 
    
28. void greenn() {
29.   int i = 0;
30.   for (t=2000; millis() > t; t=t+2150) {
31.     po[i].appear(1);
32.     i++;
33.   }
34. }
35. 
    
36. void britva() {
37.   rect(mouseX, mouseY, 56, 143);
38. }
39. 
    
40. void keyPressed() {
41.   pause = !pause;
42. }
43. 
    
44. class Head {
45.   float yloc = -50;
46.   float xloc = random(100, 680);
47.   float sel;
48.   float rcol;
49.   float gcol; 
50.   float switch1 = 1;
51.   float switch2 = 1;
52.   int fixSwitch = 1;
53.   int damSwitch = 1;
54.   int saveSwitch = 1;
55.   int missSwitch = 1;
56.   int playSwitch = 0;
57.   void appear(float Tsel) {
58.     sel = Tsel;
59.     //Selector
60.     if (sel == 1) {
61.       fill(0, 255, 0);
62.     } else if (sel == 2) {
63.       fill(255, 0, 0);
64.     }
65.     rect(xloc, yloc, 40, 57);
66.     if (!pause) {
67.       yloc=yloc+speed;
68.       // 1 --- DAMAGE
69.       if (mouseY<yloc+40 && mouseY+90>yloc && sel == 2 && xloc+30>mouseX && xloc < mouseX+30) {
70.         yloc++;
71.         rect(xloc, yloc, 40, 57);
72.         if (damSwitch==1) {
73.           damSwitch = 0;
74.         }
75.       } 
76.       // 1 --- FIX 
77.       if (mouseY<yloc+40 && mouseY+90>yloc && sel == 1 && xloc+30>mouseX && xloc < mouseX+30) {
78.         yloc+=2;
79.         rect(xloc, yloc, 40, 57);
80.         if (fixSwitch==1) {
81.           fixSwitch=0;
82.         }
83.       }
84.     }
85.   }
86. }

Thanks for reply! Sorry for a messy code. I've just cleaned it up and updated it. I removed unused variables, changed "paused" to "gamePause", removed unneeded switches. 

Indeed you've adapted your code now to most of the suggestion above, except for one thing: the switch1 boolean. Which is crucial because once it's toggled, it creates the problem you describe. You are using an or || conditional. Once your switch boolean is set to true once, it's never set to false again and your shapes will keep falling fast. I already mentioned this above and removed them from the simplified code accordingly.

Currently you change yloc in three places and you have three (overlapping!) rect() calls for each single shape. To imrpove your code I would suggest an update() method where all the yloc changes, booleans etc. are handled. A mouseCheck() method which handles the mouseover part of the code. And a display() method where you just simply display the shape based on the final location, state etc.

The thing is: switch1 doesn't have to become false again. Rectangle you catched has to fall down and never change it's speed back to normal or interact wth the player. Rectangles are overlapped because in the full version of the game there are images falling instead of rectangles, and after you catch them a new one is displayed on the top of original.