I have computers, but I prefer my phone and do most of my personal work on it. If I never have to set up a computer again... great. So that said, recently I've been getting into the Raspberry Pi. If you're playing newer games like Killer Instinct on it, then it's too slow. But for older games it works fine. So if you're going for simpsons type 4 player games then it's great. Something that uses 3-d polygons... not so much.
If you use a raspberry pi the cost won't be less than a crappy old windows machine. Everything included (power supply, Pi, shutdown switch, memory card, wi-fi dongle, HDMI-VGA adapter) will set you back about $85. The upside is that you can download a copy of the whole OS with all emulators already set up and ready to go (along with a front-end) and save it to a memory card. Plug in the memory card and you're ready to go. Then you copy the roms across via FTP or SAMBA shares. So... I'm just so sick of messing with my crappy old PCs and all of the windows nonsense. For my simple needs I love using the RPI. (Plugs easily into the VGA monitor)
1. If you go with a windows box... free to $200 somewhere in that range. Go for something above 2.2ghz
2. Ipac 2 or
GroovyGameGear Keywiz 40-st 3 I personally use low-tech stuff like Mala. Doesn't require much work to set up (though I think the themes out of the box are pretty terrible) and the processor requirements are very low. Hyper spin's a pretty big deal these days, but I don't use it. My machines are too old and crappy.
4. ? say what now? Emulator? Mame and FCEUX for Nes emulation.
5. If you want to hook the computer into your cabinets power supply and start up and shut down with a single switch you will have to consider a few things. Helps if you can use VNC over Wi-Fi to administer the machine. You'll need to download all of the Mame artwork, samples, control panels, overlays and anything else you want for your front end.