Without knowing the full extent of your situation it's hard to say. If you have full legal custody, and your child's father has none, then legally you probably don't have to let him see her.
However, if he is a decent man (as in he won't harm your daughter, and loves her), then you may want to consider letting her visit him. She has a right to see her father, and he has a right to see his daughter. Just because you both are separated shouldn't harm your daughters relationship with either of you.
If your daughter is old enough, ask her if she would like to see him. She should really be the one to make the decision providing you know your husband won't harm her in any way.
He can still file an injunction
Yes. He would have to petition the court for custody.
yes he should do
Sole custody means you have all the power. You can choose to keep the father out of their life until the children turn 18 if you want to or you can allow visits.
no, but the father could file for custody see links below
Yes, if the father is not given custody he will be obligated to keep paying support to whomever the court awards custody or guardianship of the children.
you get it for her.
How would he do that without being in possession?
yes
My answer to that would be 'No'. The father is responsible for providing child-support regardless of who has custody of the child; at least until the age of 18.
If the father has no legal custody, but is paying child support, he should be at least told. It's his child, too. The parents divorced each other - they did not divorce themselves from the child.
The father would file for a visitation order, not child support. Generally, if the parents are unmarried the mother has sole custody and control in most states until the father can establish his paternity. Remember, a child's mother can always be identified by medical records. Since the father didn't give birth and he was not legally married at the time of the birth he can establish his paternity through a DNA test. A paternity test can be arranged through the court. Once paternity is established in court, the father can request visitations or custody through the court. If the mother retains physical custody she can request that the court issue a child support order. If the father gets physical custody he can request a child support order.