You really do like your 1/8"s don't you

(Should be 5/8* versus 1/2" or 3/4", sorry I'm a stickler for these things)
Anywho's, is there any reason you with to use 5/8" for everything?
The 'general' thing to do is to use 3/4" board and 3/4" t-moulding for the cabinet sides. You can however use and board thickness you wish. A quick look a t-moulding.com shows that they stock 1/2", 9/16", 5/8", 11/16", 3/4", 13/16", 1", 1.25" and 1.5" t-moulding. (That's if you wish to use t-moulding)
You could, if you wish, use 1/2" MDF for the other panels on your cabinet. This would help keep the weight down slightly (although you Will still have a heavy cab, its suprising how heavy a few sheets of MDF can add up to).
As pcolson said, if you want to use plexi on your CP the ideal thing is to use 5/8" MDF with 1/8" plexi (6/8"

) and then use 3/4" t-moulding. You could however use any combination of board and plexi etc thickness to match and of the t-moulding thicknesses.
Paint wont be an issue, all the board thicknesses will paint the same. (you could also use vinyl or laminate, I'm going for vinyl on mine.)
'Feel' wont really be an issue either apart from the sturdiness of the cab but as long as you build the cab properly you can make it sturdy using any of the board thicknesses.
As for Soul Caliber, I couldn't tell you if it is supported in MAME (to be honest its a couple of years since I have used MAME

) but you could always put a Dreamcast i your cab and hack a couple of pads to your controls. That's what I'm going to do.