With this in mind, how do I get my trackball and spinner working?
(trackball and spinner currently work fine as a mouse)
Hmmm. Does that last phrase mean your mouse works in mame (assuming you also have a mouse), or that the TB & spinner don't work in windows? Or do you just mean mame doesn't register any mouse when you try to remap?
Although it's better to do as Paul Olson said, you can also go to the mame folder, right click, select new, text file, open new file in notepad, add "
mouse 1" (not including quotes), save, then rename the file to
mame.ini. (Make sure you aren't "hiding known extensions", though.)
Yeah, I'm having the same problem but with the spinner only. You see I spin left and the PC reads it as going right. It's really hard to play arknoid on it this way. 
Also is there a way to set the speed for the spinner 
Does "the PC sees it going right" include in windows, or just in mame? If both in windows and mame are reversed, the best thing is to fix the spinner; this might be as easy as swapping the X1 & X2 (or Xa & Xb) wires. But be sure not to swap either with the +5volt wire.

You can fix it in just mame, but... you have to do it for each game, and each game you fix will reverse any mouse/TB you want to use in that game. Start game, tab, Analog Controls, go to Dial Reverse, and change it to what it's not.
For spinner speed, it's also done in the Analog Controls menu. It's called analog sensitivity. The Digital Speed is only for simulating analog signals for digital inputs (such as keyboard or 8-way sticks).
Another note, do NOT map your spinner to Dial Left, nor Dial Right. These are for the above mentioned digital inputs, and should NOT be used if you have true analog inputs.