There is no such tournament as the British Open do you mean the Open Championship?
There are currnetly 9 golf courses in the British Open rotation, they are The Old Course St. Andrews, Carnoustie, Muirfield, Turnberry, Royal Troon, Royal Birkdale, Royal St Georges, Royal Liverpool and Royal Lytham and St Annes.
http://www.golforderonline.com
Yes, an amateur would win the Claret Jug if he won the Open Champoinship. However, he would not win the prize money, as that would contavene his amateur status
Rory McIlroy set the course record today in the first round of the 2010 British Open with a 63, 9 under par.
The ones in the current roatation are, St. Andrews, Royal Birkdale, Royal Troon, Royal St. Georges, Carnoustie, Turnberry, Royal Liverpool (Hoylake), Royal Lytham and St. Annes and Muirfield.
The courses which have held it, but are not in the current rotation are Royal Portrush, Prestwick GC, Musselbrugh Links, Prince's GC and Royal Cinque Ports GC.
I couldn't believe it either. It was down to his finish in the Order of Merit 2008.