Interesting, the original MK Midway cabinet I had did not have the slide out. Only game I've had that did so far is the Berzerk I'm working on.
Of the 3 MAME cabinets I'm currently working on, I can tell you that so far the cooling need depends (mostly) on the computer placed inside the cabinet. One PC, still in its case with side panel still on, had overheating trouble. The PC was higher end than the other two I've been using, and its video card was nicer, etc. Just like any PC setup, the better the components are and the harder you work them can dictate how much hotter things will get. Keep in mind that a properly designed computer case is usually built to cool best with the side of the case on and air flowing through as designed. But that's the "ideal" situation. They don't expect you to throw it in a big box with a nice warm CRT, etc. I've used lower-end PCs like the classic Dell Optiplex 745 setup, and those things never seem to overheat.
Fans are cheap, and wiring them up around a cabinet is pretty simple given all the space in there. I advise tossing in at least one intake (down low) and one exhaust (up high). Depending on your cabinet, you may even be able to use the existing vents and mount fans to them for flow. Maybe post a couple of pics so we can see what you're working with.
As for in the PC case vs. out of the case, there have been countless discussions on that topic on the forums. Opinions vary. I've done it both ways, and haven't had problems with either. I kind of like the "classic" arcade look of a board mounted to the wall, mounted power supply, etc. But I also have to admit that just putting a PC case right in the cabinet is a lot simpler, and easy to yank out if needed. It really seems to be personal preference. With the slide-out board there, it might be nice looking to mount everything to the board for fun.