I'll answer both your questions -
I'm not saying he does or does not have Asperger I am asking why within the court no evidence has been presented that I am aware of proving he has had this condition all his life.
The only thing which matters is whether his current condition creates an obstacle to extradition. There is no absolute need to prove he had any condition all his life to prove he suffers from it now.
People slide around the spectrum all the time. I imagine he had far fewer issues when he wasn't facing extradition than he is now. Perhaps that's his fault, maybe not, but that doesn't matter. The question is whether it is an obstacle to extradition.
If this wasn't the US government that he hacked and it was an American university computer system causing the same amount of damage/problems would people still be against his extradition?
Yes. It is not the issue of the crime or victim, it is an issue of the American justice and detention system. He would face the same risk from those in both cases.
I personally oppose all extraditions to America on the grounds that detention, justice, punishment and incarceration do not meet acceptable civilised standards.