The gearbox fault that you are referring to is the Midlands 'box fitted to all petrol Ones and Coopers built before July 2004. It's quite nice to use, but it's prone to failure and new replacements are not available as the Midland Gears factory has been closed. Diesel Ones and Cooper S's were not affected.
Buy petrol Ones and Coopers post July '04 if at all possible, or buy low mileage pre July '04 and check that the gearbox is working as it should.
Rust is not normally a problem, but lift the rubber seal that sits above the seam on top of each sill and check for rust there.
Power steering pumps on petrol MINI's before September '06 were noisy from the day they were built. Rover rejected the system originally proposed by BMW has having too little feedback and designed their own system which, as a by product was noisy; especially on full lock. With the second series cars (and with Rover no longer involved), BMW got to fit the system that they originally wanted.
As Streeters says, the Ones built up to September '06 are a superb starting point for serious tuning, especially if fitted with the post July '04 gearbox
Tigger.
PS: An easy way to tell if it's a pre July '04 build car is that the interior clock is built into a pod by the rear view mirror, if it's post July '04 it's next to the mileometer in the speedo.