So live rock can be cured or uncured, right? Can uncured live rock be cured in the aquarium? I don't have a garage to cure the rock outside. Also, if it's dry live rock, can that be cured in the aquarium too? It will be a new tank with no livestock.
Here is the way I use these terms:
Live rock - Rock with 'life', taken from the ocean. OR rock from a well established tank. This implies the rock is cured and full of life (hitch hikers).
Dry rock - Rock with no life, is dry. This could be mined or just old rock which was previously 'live' but has dried out.
Cured rock - Rock which has been through the full nitrogen cycle, has all the nitrifying bacteria on it and was once dry.
Uncured rock - implies there is no life, but could be wet I suppose (I would probably not use this term).
For example: I bought a few pieces of live rock from various sources. I also bought a lot of dry rock. I then put them together in a couple of tubs for a few months. I then took the live and cured rock out, and put them into my new tank.
With this in mind, to answer your questions in order:
I would not consider live rock cured or uncured. Live rock is live rock, cured rock really just has the nitrifying bacteria on it.
When you start a tank and go through the nitrogen cycle, you are curing everything in the tank, i.e. the dry rock and the sand/substrate.
I would not consider 'dry live rock' to be a thing, this is just dry rock. Yes, it would also be cured during the initial nitrogen cycle.