When I say: I have an AC in my bedroom, the abbreviation of AC is:
to mean the name of the machine itself.
What is the correct use and why?
And 2. air conditioning.
A room can be air conditioned (3), but it cannot have air conditioned.
Air condition (4) would mean the state of the air, rather than the device that brings the air to a satisfactory state.
The presence of the indefinite article an requires that the following word be a noun. Only choice 1 air conditioner fulfills that requirement.
A room with an air conditioner
An air conditioned room
A room with air conditioning
All the above are idiomatically correct.
We usually don't use a hyphen, but it may be necessary in some circumstances:
He was used to air-conditioned air.