The AC (air conditioner) won't run without AC (alternating current) power. Unless you had a generator running or a very, very long extension cord, it did not run. Regardless, RV air conditioners are perfectly fine running while on the road; this is done frequently in motorhomes. It could be damaged if running when your trailer is laying on its side, say, but you have more pressing concerns in that case than the air conditioner's health.
The propane detector probably beeps from low voltage. Whether or not your tow vehicle supplies a bit of power to the trailer while underway or not depends on the electrical connection between them. Unless you have a six or seven wire connection, you don't have any power to charge the battery etc. while underway. Since you say "car," I'm wildly guessing you probably have a four wire flat plug, which only operates the tail/clearance lights, the left turn/stop light, and the right turn/stop light.
Even with a six or seven wire setup, the power available from the tow vehicle is rather limited.