Being a Bugeye owner the A-series part of things will be familiar to you. The gearbox will be a bit different but it's still a BMC design and won't be too strange.
Because the Mini remained in production so long, bits can be mixed and matched over the years (with care). What remains true though is that regardless of the age of the car, fixing and painting the shell remains a singularly expensive proposition, particularly going to bare metal. Therefore, look for and avoid rust. Get the most rust-free Mini you can find. The mechanical bits are less expensive and easier to deal with. Beyond the sills look at the front floors. Lift the vinyl off the bottom of the rear companion boxes and look for rust. In the boot, look for rot along the back edge where the hinges are. All of those can be complicated repairs if the rust is severe. The rear subframe bolts to a surface just below the back seat. That area can rust and it's also expensive to fix. The rear subframe itself can rust and it has to be replaced, there are no off-the-shelf patch panels for it. The front subframe is usually well protected with oil. However, up front, look for any signs of rust on the scuttle panel in front of the windshield extending down to the front wings. That's another complicated repair. Roof pillar repairs can also be tricky. On cars with sliding door windows, look for rust along the bottom of the door frames. Expect the sills to be bad but pay attention to where the crossmember connects to the inner sills. On cars with sliding window doors, the A-panels where the hinges are on the car body are 2-layers thick. If you see rust bubbles anywhere on the A-panels they will need to be replaced as part of your bare metal respray. They aren't too hard to do but fitting them with the wings in place can be a challenge. All of the body panels can be bought somewhere or scavenged from a donor car. It's not that there's a shortage, just a lot of work. Choose a car requiring the least amount of bodywork. Quality panels come from Heritage, M-Machine, and there are still some NOS Rover parts around. The generic pattern panels have a reputation of requiring a lot of tweaking to get the fit right. Your time is valuable so getting quality panels in the end will save you time and therefore... money.
As far as the go fast bits, the 12" wheels will allow you more options for brakes. Standard Minis came with drums all around. The Cooper had 7" front disk brakes reported to be only equal to (or lesser than) the drums. The Cooper-S added 7.5" disks that were better. When the 12" wheels came about they could squeeze bigger 8.4" disks under the rims. If your would-be project has drums and you're going to change or beef up the engine you may wish to consider buying and converting to the 8.4" brakes.
Good luck with your project. Let us know what you decide to do.