I was under the impression that the Vault 101 was closed to the outside world. So how was James able to get in with his baby (you) to raise you in the vault? If the overseer agreed to let him in, why was he in such a bother when James left again?

That's probably the only thing that bothers me about the plot. From the start, a great deal is made out of the fact that the Vault 101 is shut off to the outside world, no one gets in and no one gets out. Then, as you play the game, it very casually refers to your father going into the vault when you were born, like it's some kind of an open sanctuary.

During that time, the vault was more open.

http://fallout.wikia.com/wiki/James_(Fallout_3)

http://fallout.gamepedia.com/Vault_101#Background. The Vault 101 experiment only makes sense if you read the description provided in the strategy guide (which also provides key parts of the backstory, instead of chucking a few scraps of information at the player, expecting them to reconstruct the whole history of the Vault on their own).