Golfers get paid quite a bit for winning a tournament, but where does the money come from and who determines who-gets-what.
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Behold the prize money list for the 2006 US Masters.
Quick calculations reveal that the top five players were to receive respectively 18.0%, 10.8%, 6.8%, 4.8%, and 4.0% of the total purse. Looking around randomly elsewhere, it seems that 18.0% of the pot is par for the course for a tournament victor. For example, the 2003 US Open at Olympia Fields carried a total prize fund of $6,000,000, of which the winner trousered 18.0%, or $1,080,000 in hard cash. I’d be surprised if other tournaments didn’t use the same percentage basis for their rewards.
While the Augusta National allowed for 47 players making the cut in 2006, the actual number would have depended on where the cut was applied. I don’t know exactly how many players made the final 36 holes that year, but the point is that the percentages at the lower end of the money list would have varied slightly from the 0.3% projected percentage to be earned by the 47th finisher, if more or less than 47 competitors played the weekend.
As to the source of all this money, I suggest that TV broadcasting rights provide the bulk of it, with gate receipts making a lesser contribution. Of course, where a tournament is sponsored then some of the cash will come from the marketing budget of the companies concerned.
Do the players have to pay a buy in fee? Whether or not the sponsor actually covers it.
Where a player is exempt from tournament qualification because he is a previous champion, or his world ranking is high enough, or his position on the money list merits it, or his performances in other tournaments are good enough, the answer is elusive. 
However, using the British and US Opens as examples, anyone can enter these competitions as long as they are accredited professionals or amateurs and have a scratch handicap (or a handicap index of 1.4 or lower for the US Open). In such cases, the entry form must be accompanied by a fee of £125 for the British and $150 for the US Open.