Built-in generators are generally a good bit quieter than most open-frame generators, though it does vary somewhat with the built-in ones depending on the particular installation. Some are very quiet, some are kinda noisy. A Honda (or other) inverter generator is often quieter than many built-in generators, especially the lower-end built in ones (i.e. virtually every genset on a gas powered motorhome) and especially at modest load levels.
Using a generator is a part of dry camping, but so is adhering to campground rules. Yes, they could and quite possibly might kick you out if you persist in using a too-loud generator, just as if you were blasting music during quiet hours or draining sewage on the ground or cutting tree limbs without proper permission.
The 1200W generator will not run a 1000W microwave satisfactorily. It should be adequate for charging your batteries, depending on what sort of converter you have, but you'd be quite limited otherwise—and doubly so if you want to charge the batteries and do other things requiring much electricity at the same time. If you have an inverter capable of running the microwave, you might be able to get by with a sort of double conversion, using the generator to charge the batteries while running the microwave off the inverter, assuming you don't need particularly long microwave runtimes. It would require some careful arrangement of the wiring to pull off.
