Arcade Machine = what you are building
Arcade = a location with many arcade machines
Sorry...pet peeve...
Anyway... do yourself a favor and add a single exhaust fan toward the top OR if there is venting in place at the top add an intake fan at the bottom. Simply add a grill if you're worried about the cats.
I can assure you that cabinet gets hot...when I open my CP after my MAME rig has been running for a while I can feel the heat that comes out from the CP and that is with a fan running.
EDIT: BTW, you plan on adding a couple fans to your PC case that is going to be sitting INSIDE your cabinet?

Seems sort of counter productive. Why blow hot air from inside the cabinet in to an enclosed case? Take those fans, mount them to the cabinet door or one as an intake through the cabinet door blowing toward the PC and one as an exhaust at the top of the cab...then take the sides off the PC case. All you need the case for is the mounting locations for the PCB, HD, etc....no need to have the sides on...
PERSONALLY, I prefer mounting my computers for mame rigs to a piece of plywood that gets mounted to the side or bottom of the cab....it makes for real easy removal and provides maximum airflow.