short in overhead lights, porch light & a.c.

Wire nuts are a source of a lot of electrical problems. Might check the wire nuts in the problem wiring circuit.
Don

When strange things happen to multiple devices like that, I always first suspect a bad ground somewhere.

I would suspect a bad ground connection, common to all the circuits in question. Finding the ground...that could be a challenge. Maybe the maker can supply you with a wiring diagram if you're fortunate.

normally what is called a 'short circuit' means a power path to ground and blown fuses

turning on a light and having the A/C shut off means a 'Voltage' problem on the circuit feeding the thermostat
could be the TS control power is coming off the same circuit as that light switch

how long have you had this RV, is this a NEW problem
check the batteries and the converter for loose or dirty connections

a recent purchase, that the previous owner updated with a newer A/C and TS
then something might be connected WRONG

Time for the digital multi-meter...

You have a wiring issue. If I was to guess, one light switch is interrupting power from the thermostat. A further guess would be that some of the lights are interrupted by switches using the (-) phase of the 12 V wiring instead of the correct phasing, using the (+) phase.

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