#1
I just want to sort some theory out for my song.

For the start it uses various C chords, G, Am.

Then it goes into alternating between F7add13 & F7add12, now I don't know if the notation there is right.

F Lydian can be used of that yes?
#2
F7add13 is written F13. A 12 chord doesn't exist, as the 12 is simply the octave of the 5 which is already there, so you would write that as just F7.

Assuming you don't mean maj7, you wouldn't use F Lydian there. You would use F Mixolydian, as Mixo has the lowered seventh. Yes, it goes out of the C major scale, but that doesn't matter if it sounds good.