What happens to the winnings of amateurs such as Michelle Wei and Morgan Pressel ?
Do they just not get the money at all, or is the money put away in some sort of trust fund that they can collect on at some future time?
Would the answer be different if they weren’t minors?
Is the answer different for amateurs in other (American) sports such as tennis?
If Michelle Wie wins today, the prize money for first place goes to the top placing professional, even if they are the third or fourth place finisher.
Amatuers can’t accept any prizemoney at all, even deferred.
I think I can answer as far as golf is concerned. There are no such things as amateur winnings. If Michelle won the tournament, she might get the trophy but the first prize money goes to whoever comes in second.
The same applies to whatever prize money she is entitled to. It goes to the next winning score.
I remember a story about an amateur in one of the Buick tournaments(they sponsor a couple) a few years back. He got a hole-in-one and won a brand new Buick. He was faced with the prospect of accepting the car and in so doing, declaring himself a professional or declining the car and retaining his amateur status. He accepted the car as he was a college student and had no professional aspirations and it would affect him not at all. He lost his amateur status but drove back to school in a brand new car. Pretty cool!
There’s not much doubt Michelle is going to turn professional and if she was to win a major tournament the temptation to accept would be strong. The prize money on the LPGA is not as high as on the mens tour but not chump change either.
Don’t know about other sports but the principle is probably the same. 
so what is the disadvatage of turning pro? Does that mean you cant play in most minor tournaments?
It means you can’t play in any amatuer tournaments, from the U.S. amatuer to your local club’s championship. It also means you can’t accept a college scholarship (in golf).
Where are the rules for these things specified?
It isn’t that I don’t believe the answers, but the person with whom I’ve argued this isn’t going to take “I read it on a message board” as a definitive answer.
The NCAA sets the rules for US college athletcs. Possibly the AAU sets the rules for non-college amateurs. The United States Golf Associations sets the rules for golfers.