In Westeros the line of succession goes like it goes in most medieval European monarchies- by primogeniture.
So, 1. in line -ELDEST son (in this case Rhaegar)
If he dies childless- the crown goes to his younger brothers and sisters (viserys, than Dany)
If he has living children, as we are led to believe, they take precedence over their aunts and uncles. So that makes Aegon first in line for the Iron Throne, if he is really Rhaegar's son (methinks not, btw.)
That's the whole reason for Stannis' claim on the throne- Joff, Tommen and the rest not being Robert's children, so he would be Robert's heir then.
You're wrong. Primogeniture means the eldest son still aliveat the death of the king. When Rhaegar died before his father it made his brother Viserys the heir of the kingdom. But when Viserys died, Aegon had a claim as strong as Daenerys to be the rightful heir. The first Dance of Dragons did not resolve if women can be heir by themselves.
If Jon Snow is the son of Rhaegar and Lyanna he is still a Snow and not a Blakfyre. Blakfyre was the name chosen by pure blood Tragaryens bastards. Other Targaryens bastards take thebastard's name from their motherland Like Bittersteel and Bloodraven both from Riverland mothers and both named Rivers. For Lyanna Stark it is Snow.