What engines am I limited to on this car? I know I can do a direct swap to the 3.8 SC motor but I'm wondering if a 5.0 or the likes could be shoe horned in without too many problems.
The 5.0 was an option on 91-93 cars, so it'd definitely fit. You'd need one from a T-Bird or Cougar, or a 1994-1995 Mustang, though, as they have a lower-profile intake manifold to clear the hood. You could also go with a 4.6 SOHC (were optional in 94-97 models) or a 4.6 DOHC (standard in Lincoln Mark VIII - similar chassis as your MN12 Cougar). Regardless of which engine swap you go with it'd be better to have the whole donor car so you could get the wiring, mounts, and any other parts you may need...
I'm thinking that the DOHC swap would be a decent swap but would I need to swap the computer as well and have to re-program it somehow for my car.
If I try to go with a 5.0 that uses the 4R70W tranny is it still a viable option if I go with a new hood that has the 2.5" cowl to it?
Regardless of your engine choice, you'll need the computer (and wiring) for that engine. Your stock 3.8 computer won't run a V8, and very likely wouldn't even run a supercharged V6, since your '90 would have speed density.
I dunno if any 5.0 ever did come with the 4R70W (I think they only came with AOD's and AOD-E's, since the 5.0 went out of production in cars in 1995). You could go with a 5.0 without the "elbow" with a cowl hood (heck, it might even barely fit under a stock hood), but it'd probably be cheaper to find a T-Bird or 94-95 Mustang intake...
I have one of those intakes, if you ever get your engine swap off of the ground.