A lot of questions in your post
I think in three weeks you have time for making that trip without stress. You may even visit more places.
In regard of the viewing of the Pyrenees I believe you are going to get a good viewing of them. I will suggest to visit some other places as Aigüestortes, Val d'Aran,... if you can (but Ordesa and Andorra are quite beautiful).
School will start in mid-September, so that's no problem.
There is a camping in El Escorial, at the foot of the mountain. That town is sooooo beautiful... Don't miss the herrerian monumensts: The monastery and the San Bernabe's Church.
Regards.