I moved a metal electrical box for a smoke detector from a drop ceiling to a drywall ceiling a couple feet away. This box has a doorbell transformer attached to it. In order to mount the box, the transformer will be inside the ceiling. Is this a fire hazard?

This is not a legal installation. The Code does not allow you to 'bury' the transformer within the drywall; it must be left accessible at all times.

I want to run 3 individual circuits in the same conduit. I’m going to install 3 hots and 3 neutrals instead of 3 hots and 1 neutral because I want individual breakers for each circuit. Is this considered three or six current carrying conductors if I do it this way?

Both neutral (grounded) and 'hot' (ungrounded) conductors are all considered current carrying conductors. The grounding wire(s) that must also be installed in the same raceway is not considered current carrying for derating purposes, but must be counted for proper conduit sizing.
As a side note, if you wanted to run a multi-conductor circuit, you can't have more than two hot conductors sharing a neutral conductor (unless it’s a 3-phase circuit), and those hot conductors must be installed on opposing phases.

Why is the wall warm next to one of my outlets? Should this be checked and if so, what would an electrician be looking for?

You should definitely have this checked by a licensed electrician. It may be a loose connection on an overloaded circuit.

Can two different branch circuits originating from two different panels be run together in the same branch conduit? They are both of the same voltage. They are both motor circuits. We have a tight situation with getting down through a wall on a remodel, and combining the two circuits together in the same conduit would ease the problem.

You should be OK with that; however, the source of each circuit should be clearly identified at the disconnect location near the equipment being powered. This is a Code requirement.