The answer to all of your questions is yes. LP detector is a must have. Ours gets tested every time the dog sleeps in front of it.
They make combo CO/Smoke detectors for RV's, supposedly they are made differently from regular house ones. However I suspect either type will work just fine. I would not be without either one. Seems every year a few people die in an RV somewhere due to a faulty furnace, using a stove top/oven for heating or a generator that allow CO to seep into the RV. I know we had the CO detector in our unit go off one time due to the clown behind us running his generator, we had our window on that side partially open with a roof vent running and pulled the exhaust in. We were outside at the time.