("I had an "authorized repairman" service it")
There are no such things as Authorized Repairmen for the Arcade Legends machines. That said, anyone familiar with Arcade Legends or UltraCade machine repair can work on it.
If your machine has a white case, then it was a computer that was built by Chicago Gaming and not by UltraCade Technologies. Chicago Gaming was authorized to build the computers inside. They mostly used DELL motherboards.
The shutting off when hitting the exit button was a grounding problem either at the button or on the I/O card, and had nothing to do with your computer.
("I have three questions for you guys and gals, if you would be so kind: First, do you think I really need a new hard-drive and if so, can I install it myself if I do not know much about computers?")
It's a standard PC hard drive, however you will need the unlock code to re-install the software.
("Second, if I re-install the games I added-on later, like ones in the Mega Pack, the Sports Pack, etc., will I have the same problem with the "Exit" button as before? If so, does it really matter that this happens or is damage being done to the game when it does?")
No, the games have nothing to do with your Exit button problem. You can just plug in your game packs and they will reinstall onto your system.
("Lastly, and this is going to sound like a stupid question, but where the heck is the battery compartment for the CR-2032 battery? ")
There is no compartment, it's on the motherboard of the computer.