An 1182 x 660 attachment?!? GREAT SCOTT MAN! Seriously though, please trim that down. My monitor is at 1280 x 960 and I
still have to scroll back-and-forth to read your post.
Anyways, to answer your questions...
I don't quite understand what all the buttons are.
what are the 3 sets of buttons on the middle row used for?
I guessed that the top row on the right is a set of 4 buttons. Are these player 1-4 buttons?
What would the other 5 buttons on the top row be used for?
"Trackball" buttons? Maybe "admin" buttons like Pause, Quit, Tab, etc.? In all honesty though,
if you don't know what they are, YOU DON'T NEED THEM. I only have two "admin" buttons (that is, buttons that aren't "gameplay") on my panel: Pause and Quit. Some people like to have a "credit" button next to each Player Start button, so maybe those. But think about what you want to have - don't just slap a metric ton of buttons on there unless you actually have a good reason to do so.
What is the opinion of this coin door?
http://www.happcontrols.com/coindoors/40000800.htm
Happcontrols.com isn't coming up for me right now, so I dunno. Most people go with the "Over/Under" door from Happ though.
I want to set it up so that when you press the coin return button it gives that player a credit.
You'll have to wire in a switch yourself. 1UP did this and
documented how he did so on his website.
What kind of buttons are prefered? Where is the best place to buy them?
The Happ Pushbutton with Horizontal Microswitch is probably THE standard. I'd link ya to it, but like I said, I can't get to happcontrols.com for some reason.
Do I want joysticks that I can switch from 8 way to 4 way?
I don't know...do you?

Playing 4-way-only games like Pac Man, Frogger, Donkey Kong, etc. with an 8-way joystick produces less-than-desireable results (the diagonal directions can screw up gameplay). That said though, even if you DO want to play 4-way-only games, I can't think of a single game where you'd need MULTIPLE 4-way joysticks. If you do play a lot of 4-way-only games, I would think about mounting a separate 4-way joystick on your control panel personally. Either that, or maybe just have the Player 1 (and maybe the Player 2) joystick be switchable. For Player 3 and Player 4, it won't matter because you'll never use them as 4-way sticks.
What about the T-stik plus from ultimarc? What are some better alternatives? Where should I buy them from?
This is like asking "What's the best car, Chevy or Ford?"...you're going to get different responses based on people's opinions. I haven't use the T-Stik Plus myself, but I've heard good things about it.
What trackballs are recomended?
The Happ Trackball - while probably the most expensive - seems to be the best in terms of feel AND durability. I own one and I absolutely love it. I don't think that any of the other trackballs "suck" though. Again, it's my personal opinion.
What about spinners?
Opinion again, but I'd definitely go with the OSCAR Controls Vortex spinner. It's the most "arcade authentic" spinner out there.
*WHEW!* Hope this helps!