No, you do not have to be able to afford the quest in order to select it. I did the first ambition without any cheats, and by the end I only had 2 QP for a list of 3-4 QP quests. Still, I had no problem selecting from any quest on the list. The QP only exist to keep track of time, so to speak. It doesn't matter if you hit zero or negative points, but when you do, the ambition is over.