Courts never make the father of the child pay for "preparing" for the child at the mother's residence. If you want to spend your money, getting things ready at your house, you can. For all you know, he is spending his money to get his house ready for the newborn also.
When the child is born, you will file with family court and until the court date, neither mom or dad has more legal access to the child. The courts will decide physical custody and visitation rights, if needed. The courts will also determine child support, if any.
" He will not be a good influence at all and I m scared of how differently she will turn out if he is in her life a lot" = he's not good enough for your child to be part of his life, but he was good enough for you to the hanky panky with underessed? Your standards for YOU should be as high as your standards for your child! Seems like the child is the one that got the short end of the stick with both parents.