Right now, I wear Asics 2 in 1 running shorts. The inner liner is sort of a boxer brief style and is very thin and fairly snug fitting. Before I found these shorts, I had chafing issues on my upper thighs. Once I put these on, I never had any chafing again. They also have no zippers, so there is nothing to rub strangely under a hipbelt.
To answer your questions:
1. Yes, I wash them out daily. If I think I can do so in an environmentally sound manner, I will rinse them directly in a stream or lake. Otherwise (like if I have applied sunblock or deet to myself before or while wearing the shorts), I will bring a wash basin made of a cut-up Reliance water jug. Then I gather water and rinse the shorts out away from the water source. In any event, I don't use any soap. I bring 2 pairs of shorts so I can be wearing one pair while the other dries hanging from my pack. Then, before turning in for the night, I change into my "clean" pair. In the morning, I am already dressed for hiking, and can get on the trail right away.
2. Since the shorts have a liner, wearing long underwear or tights underneath isn't possible. I put light pants (BPL Thorofare) on top of the shorts. Works well for anything other than winter.
3. I think running shorts are generally lighter, better at moisture management, and better at reducing chafing, than swimsuits, but I am sure it depends on the specific model of each that you're comparing.
Hope it helps, and happy trails!