Increasing the diameter of the pipe should supply greater volume, but it won't do anything to the pressure...don't confuse pressure with volume. I'd check the actual pressure you have - HD and any hardware store, sells a water pressure gauge. It's the only real way to know what you have now. Because it can freeze there, you want to make sure the pipe is run as close to the floor with insulation under it so that it traps some of the trailer's heat. Pex won't typically split if it freezes, but fittings can. Remember, insulation doesn't create heat, it just traps it. If there isn't any, the pipe will still freeze.
3/4" pex is about the same ID as 1/2" copper, so yes, I'd run 3/4" pex to the bathroom group. The tub, I'd convert 3/4" to the likely 1/2" valve you have. Everywhere else, things are flow restricted, so 1/2" is plenty. Well, a washing machine might use the 3/4" up to it as well as a hose bib.
Yes, larger pipe will hold more water, and let it take longer to get hot at the end (a recirc system might work out for you if things actually have to go that full 70'). But, as I said, 3/4" pex isn't much larger than 1/2" copper.