No, the GA-7VAX does not have onboard video so a vid card will have to be added. Nothing much was mentioned about the cab itself (monitor, etc.) so it's hard to make a recommendation. Gigabyte are great boards but in your case, make sure it's not earlier than revision 1.1. Before this version, there was no failsafe mechanism for CPU fan failure.
Your 333Mhz memory will be ok but you could use 266 without any performance hit. The motherboard supports 333Mhz memory but the XP 1800+ has a front side bus of only 266Mhz. In other words, your memory and motherboard bus are limited by the performance of the CPU. 333Mhz would be an OK idea just the same in case you upgrade the CPU sometime in the future and the price difference is negligible.
The MoBo has built in audio and it is quite decent. In fact, it's probably better than the SB PCI 128. For playing classic MAME stuff and even for listening to music, it will do very well unless you are very, very picky. Save yourself a few bucks and see how you like it first.
Lite-On 16X DVD Retail - Do you plan on watching DVD's on your cab?
Enlight cases. I may start a war here but I don't like them a lot. This is only a personal opinion but one based on a lot of experience. I've had trouble with the power supplies more times than I care to remember. If I were you (but I'm not), I would spend a few extra bucks and get an ANTEC power supply.
And why bother with a case at all? Get a good PS and mount everything directly inside the cab. It will give you better access and better airflow - something to consider when running an AMD processor. Those suckers run hot. To me, it makes little sense to run a box inside a box especially if you are talking about a full-sized cab. Make sure the cab itself has good airflow and you're set.
OS - If this is a dedicated MAME cab and you plan to use a front end, 98SE is probably your best bet. Why? Because you probably already have a copy or at least know someone who does. No extra $$$ to plop down on XP. Go to the examples page and check out MAGMA (It's in the Desktop Controller section) for more on this. It's pretty comprehensive.
In summary, yes your system will work. If it's your first time building a PC, take your time and read all of your documentation carefully and you should be OK. I believe that Tom's hardware Guide (
http://www.tomshardware.com) has a step-by-step builder's guide that you can review. Good luck.