Hi Mike,
Start by visiting and registering for the message board at Mini Mania.
https://www.minimania.com/msgthreads.cfm
There is a thread pinned to the top of the message board that discusses things to look at/watch out for when buying a Mini. Mini Mania, Mini City, and a few club sites around the U.S. have "for sale" pages with cars and parts.
As for an "honest Mini Cooper", there are Minis, Mini Coopers, and Cooper-S cars with increasing cost/value respectively. In the U.S. Minis in only the Mk1 and Mk2 configurations were originally imported here. Mk3 cars, with the first roll-up windows, are also fairly common. Beyond that you get into the cars you describe which are either re-vins (illegal) or re-shells which are legal... but questionable.
You can read up in detail on this at MM. Succinctly, the re-vins are the transfer of an older VIN plate and its title to a much newer car for tax and importation purposes. A re-shell is the installation of a new shell/body to an older car to make it mechanically sound. The truth be told... the re-shells often (but not always) involve the same swap described for a re-vin. That said, roll-up windows cars with their more comfortable interiors and better ventilation can be desirable if you live where it's hot or humid. However, going to a year 2000 or so car with it's fuel injection and electronics is a gamble as service "over here" is going to be hard to find.