There is plenty of info online around what to look out for with minis, example here from Vic min club
http://www.mini.org.au/buying-a-classic-mini maybe try contacting you local mini enthusiasts club and go tot heir next meeting. Often members have cars for sale and are always happy to talk to like minded enthusiasts.
As mentioned above, $10K should get you a reasonable round nose but will not be enough for a show winner or the more sought after cooper and cooper S unless you want to do a restoration. If you don't mind the look of the clubman they are generally cheaper than the round nose, so $10K should get you something with very little work needed. Have a think about how you want to use it, what you want from it etc. A car that is nice to cruise on a sunny Sunday is going to be very different from something you can Hill climb or do motokana in.