Is it permissible for a woman to receive the reward for building a mosque with a man's money on her behalf, and would that suffice instead of building a mosque with her own money?

It is permissible for a Muslim to dedicate the reward of a righteous deed he performed to his Muslim brother, whether alive or dead, and the reward will be established for him, Allah willing. If a person builds a mosque from his private funds and dedicates its reward to you, then the reward will be established for you. This does not prevent you from building a mosque yourself or performing a good deed from your private funds to increase your reward, and if you content yourself with what the benefactor has done, there is no sin upon you.