#1
I've got a Schecter c-1 with a floyd rose tremelo. I've noticed that the low E string is sharp when fretted / flat when open (all frets are sharp; octave between low E fretted 3rd and open G are sharp, same with low E 12th and open high E!). All other strings are fine. Using .11 gauge strings, low action.

How do I fix this? I'm thinking I have to adjust the nut, buts its a locked nut. Any ideas?
#2
Adjust the saddles until the harmonic above the 12th fret is perfectly in tune with the 12th fret.