Until/unless you are determined to be the child's father, no.
No
A childs mother must file a petition for child support in the local family court.
You can, but eventually the court will question what you are doing.
No, child support can only be taken from the birth parents.
I suggest you contact your State's child support agency about this.
Welfare
no it goes to the mother.
No, it will have no effect.
An obligor who fails to pay child support pursuant to an order can be put in jail.
It can mean something or nothing to you. It all depends if you and your ex girlfriends father were close or if you and your ex girlfriend were close and you just want to be there for support. If not then it shouldn't really matter to you.
The parents of the deceased father (the childs grandparents) can do a paternity test.
If you have gone to family court to get an order for child support you will. Don't wait for the father to volunteer to give you money. Get it in writing through the courts and if he doesn't pay he can be arrested.