A compressor will not necessarily be ruined running on low voltage, depending on how low the voltage is. The digital controlled marine systems that I sell, install and service do not shut down on a low voltage fault until it senses 103 volts for three minutes, so evidently, Dometic is not real paranoid about it. That said, I wouldn't recommend running one for an extended period on low voltage,
Any 15K BTU rooftop unit should run fine on a 20A circuit provided there are no addition draws. That's what is in the trailer. A 15A source could be iffy and you could get nuisance trips at the breaker panel. I had a marine unit that was an OEM installation that drove me nuts until I installed a 20A breaker.
If you have to use an extension cord, be sure it is constructed with #12 wire and keep it at 25'. if you have to go to 50', I would recommend that you go a size larger on the wire.