1) You should be able to. Some agents are better than others at interlining bags on long journeys, but in my experience CX agents tend to be good at it. I don't have experience with CX agents specifically at SFO though.

2) If the SFO agents tag the bags through, they will have an F tag on them. But OW carriers tend not to be really good at recognizing each others' priority tags, so don't expect much. Even so, worst case it's going to take an extra 5 min to come out in PER, surely not the end of the world after traveling >24hr and halfway around the world.

3) Assuming the answer to question (1) is yes, then you can stay airside in SIN to use the transit hotel. They will not make you dress up either.