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What kind of microwave, and what kind of power are we talking about? Will it work? Yes for a while. Will you have problems if you try using multiple appliances at once on that circuit? Most certainly yes. 12-2 is not much more expensive than 14-2, especially for a small job like this, run it in 12 and never have to worry about it.
My microwave is considered small at 800 Watts, but it has a built in convection oven. When I am running it at full tilt boogie, it will and has in the past tripped the circuit breaker when I had the ol hot pan on the same plug.
Many house type trailers have dual interior lights. They work on 120vac and 12vdc. I'm thinking perhaps something going on with that maybe? Also, when reading voltage with a voltmeter there is VERY LITTLE current being drawn so you can easily get a reading with either a bad connection on the plus or ground side. Once you add a good load a bad connection will show its face. Try this, one one interior light check the voltage at the same time another light is turned on and see if you have a massive Vdrop on the line.