One thing about a Mini auto - I've never driven one myself but believe that they can be driven a bit like a manual i.e. you can manually shift through 4 (?) gears. It is possible to swap out the auto for a manual box, with a change to the front subframe, although the autos have their enthusiasts (and their own forum too, I believe)
The advice on watching out for a new paint job is good advice. I'd look out for something 'honest' so at least you can see what may need to be done in the future.
In terms of the underbody, originally the car would be painted, but most people will have put something on to try to stave off the rust and/or repainted with a newer, potentially better product.
Diskloks are reputed to be the best protection of that kind, but you could also get an alarm and immobiliser fitted. Most people on here will have some combination of the various security devices, especially if the car is parked outdoors.
As for leaving the car outdoors: yes, they do rust, but the real enemy is road salt. In winter you'll need to keep on top of washing the car and using plenty of waxes and polishes. There are also products you can get to inject into all the nooks and crannies and cavities to keep the rust at bay e.g. Dynax, Waxoyl, etc. Fundamentally, however, the car rusts like all 1950s/60s cars rusted, in that there are water traps all over, it wasn't designed not to rust, and the construction and paint weren't as good as they are now.
Of the two links, the green car's probably had a respray (it's lost its Mayfair decals and wouldn't have had a white roof let alone a Webasto), the blue car probably hasn't. Personally I prefer the blue one if only because it's low mileage, seems fairly original (down to the wheel trims!) and has got lots of history with it. The MOT history online for it also looks OK. There are also those who reckon the build quality deteriorated during the late 1980s and into the 1990s. It is however rusting behind the rear bumper, and check the sills carefully to make sure they have the open vents/flutes. (that advice goes for any Mini)