Chapter 8, specifically section 8.8 of Audin-Damian's book on Morse Theory and Floer Homology might be helpful. In general, whenever I didn't understand something in Salamon's notes, Audin-Damian has been an excellent reference.

Indeed my intention was to have $C$ go to $C'$, thanks for the example! I guess for such a local result, the assumption should incorporate the link information. Say for example $C$ and $C'$ have the same Legendrian isotopy class of the link near the conical point, would it be possible to map $C$ to $C'$ ?

Ah, this seems very nice! I feel at this stage there should be some Moser-esque argument to change to symplectomorphism.

Has the $n=1$ been written down in literature?

have you seen the compactness chapter in McDuff Salamon book ?

for the question, yes, you can remove singularities iteratively. Removal of singularities is a local phenomenon, if you have a punctured disc with finite energy then you can fill it in .