Its assumed if there is Kiddushin, there is biah
Look, we are way beyond the discussion here, but ok. Someone suggested that perhaps because the marriage is illegal, the state would say the money doesn’t belong to him, which might invalidate the marriage. I pointed out that that is not how the law works. It is irrelevant if other things are illegal.
1. Why can a father marry off his daughter like this, but a mother cannot?
Because G-d said so. He gets to make rules, because He created the world.
Also, a mother together with a brother could if the father is dead. The marriage is d’rabbanan, and the daughter is allowed to object when she grows up.
2. Why can a father marry off his daughter like this, but cannot marry off his son like this?
Same answer.