I recommend you Chapter 5 (page 150+) of the AdS Bible,
Concerning your individual questions, which are mostly answered at the beginning of that Chapter,
This post has been migrated from (A51.SE)
the additional Virasoro generators correspond to bulk coordinate reparametrizations that preserve the metric at infinity, but they do map the ground state to excited states
yes, the CFTs in AdS/CFT typically have a nonzero central charge which is directly related to the $AdS_3$ curvature radius in the Planck units; there is no reason for $c=0$ here because the boundary CFT isn't really coupled to gravity (which is what the world sheet CFT is doing)
for the same reason, you can't directly interpret the CFT as string theory; the full string theory needs $c=0$ in total, so extra ghosts must be added; also, the interpretation of "winding/twisted" sectors is different in boundary CFTs and string CFTs. Of course, this doesn't eliminate the fact that similar "building blocks of CFTs" are used in both kinds of CFTs...