I am sorry for everything with her brother, but you have to understand some things.
She was free to move away while pregnant, She did not need your permission to do so. Since the baby was born in FL and they still live there, any court orders you file for and seek, must be filed in FL in the county they reside. It is HIGHLY unlikely you get transportation awarded. The child was born in FL, you lived in FL and you moved back. Transportation is typically only awarded when both parents reside in the same area, child is born there, then one parent moves away causing the other parent to encur costs for visitation that did not exist to begin with.
Regarding "If she lies to the court about the child support I have been paying and I am able to proof she is lying will that help, will she possibly be charged with perjury?" Don't even waste a breath thinking this will matter. She can claim you have not supported the child, you can show your receipts. It would be up to the judge and perjury just doesn't happen in family court. Child support by court order will only begin the date someone files for it. FL may go back 24 months. Because you are not court ordered to pay, you can stop for now. If you continue to pay, clearly mark any check or money order you send as CHILD SUPPORT for Ava. You would not legally get in trouble if you do not send child support because you have not been court ordered to do so. Keep in mind, if you don't help, the child goes without. That is your decision. Once someone files for child support, it will be retroactive to the date it was filed for. For example, if she filed today for child support but it does not get to court for six months, the court will order child support effective today so you will owe for those six months. So the best thing to do is put money away during those few months so you can pay it once ordered and don't start off in the negative. If you do not pay it when ordered, they will take what you owe for those six or so months, and add it to whatever amount is ordered to catch up those six months. For example, if they order you to pay 500 a month and there is 6 months till court, you would be 3000 owed for retro. They would order you to pay the 500 a month + another 50 to pay off those six months you did not pay until that 3000 is paid off.
Regarding "And Florida law I heard no longer has "non-custodial and custodial" parents in order to try to equal the parents times and rights with the child which is why I feel I might have a slight chance of transportation costs being helped in order to make sure there is a relationship established between myself and my daughter." Even if you lived in FL, you would not get equal time. With you being in PA, this is definitely not happening. You could get shared legal custody, and not always with the distance involved, and you would get visitation. Visitation is likely to be small increments of time and will increase. Eventually you will be on a long distance visitation plan which typically is long holiday weekends, rotating holidays, and part of summer. You will very likley have to pay all transportation costs.
Your name on the birth certificate (is it?) in FL does not give you any rights because you are not married. You have to file for them. Until you do, she is considered to be the sole legal custodial parent and as such, makes all decisions for the child including when and if you see the child.
You are going to have to either return to FL to file to establish paternity, visitation, and child support or hire a lawyer to do this for you.
And ignore her brother. If he calls you, hang up. If you call her and he gets on the phone, hang up. He is being a protective brother. Not saying it is called for or not, but he is not someone who is going to disappear so you are going to have to figure out how to deal with him. The court could order he is not present for exchanges, etc... but if he is, you are going to have to file to have that enforced and depending on the story they give and how you are painted, the court may not even order it.