No, the front case stays in place. Overall, the swap is simple. You do have to pull most of the bolts out of the lower timing cover though. There is one bolt on the OFH hidden behind the cover. You don't have to completely remove the timing cover, just loosen it to the point that it pulls away from the OFH enough to clear that one bolt. It's pretty easy, actually.
The more difficult part is mounting the oil cooler, although, it's also pretty easy. I didn't even have to remove the bumper to do it. I can post pictures of how/where I mounted mine if you're serious about the swap. I still have A/C too. With the cooler in its current position though, I would need to relocate it if I were to add a FMIC. I have another 90 OFH and cooler setup if you're interested. I'll sell it to you for exactly what I paid for it. PM me if you want it.
As far as running an air cooled housing with no cooler, it's not unheard of. I actually have a block off kit from STM as a backup, allowing you to do exactly that. Would it hurt anything? Probably not. But I'd expect you may loose some longevity that would be earned by lower and more controlled engine oil temps though.
For a stock daily driver that doesn't get flogged, no oil cooler would probably be acceptable. You should consider other factors, like allowing the car extra time to cool down before turning off the engine, especially if the motor has been operated hard or at a high rate of speed.
For a modded vehicle that's driven hard even on occasion, I'd say bad idea. But, when comparing no oil cooler, to an oil cooler that has the potential to kill your motor in a matter of seconds, and I'd go with no oil cooler.
