This would most likely reduce the number of people who would screen themselves. I wouldn't undergo a possibly horrible medical procedure for $100,000, but I might do it to save someone's life and feel like a good person.
I'd be surprised if that were true; you could always refuse the money and feel like an even better person (or at least just as good). And then it would attract the people who do want/need the money on top of that.
That argument makes sense only in a fully logical environment. The whole body of research into market norms and social norms shows that people are not fully logical in such contexts.
Would it be possible for someone interested in the money to get tested and ask for money in case of a match? Although I suppose the moral pressure would be quite high if you are already tested.
I suppose it's possible to ask for every penny someone has if you're a match, since they'll obviously pay you their last dime. It's an asshole move to go after someone at their most vulnerable, though, and if it became public would likely get you justifiably ostracized from any decent community. It's also currently illegal in the US.
Of course, asking your medical costs, recuperating costs, etc. to be covered is totally reasonable.
For further reading, look into social norms vs. market norms, especially the example of the Israeli day care: http://papers.ssrn.com/sol3/papers.cfm?abstract_id=180117 (paywall)