I don't know how close the ever got, but suppose Chelsea were to fall in love with Prince William. They are the same age. Even though she had never been married, would it hurt his succession?
Would such a marriage hurt or help Hilary's 2008 campaign? If Chelsea becomes a princess, I don't think she could run for office in the US
I'm pretty sure that Princess Chelsea would either need the approval of Congress or to renounce her citizenship. There again, there could be legal fiddles that avoid the need for her to take the title, see for instance the Duchess of Cornwall who is supposedly to remain a mere Duchess even as her husband becomes King.
Contrariwise, there's nothing in particular that prohibits the heir to the British throne marrying a foreigner. It's been done many times before, and a President's daughter could be considered the closest thing the colonials
p) have to offer to royalty.
The American reaction would, I think, be rather entertaining, but I'm not well placed to comment on what form that might take.