Honestly, you should probably ask a doctor about most of those questions. I've always thought syphilis is very contagious, but it might be more infectious in the early and late stages based on what you said.
I do know for a fact that there's nothing like PreP that prevents the infection like for HIV. I'm pretty sure you're much less likely to get or give the infection when you don't cum inside though.
I would think that soaking in a hot bath with ground oats, some powdered milk, and a little honey could probably help with the rash and make your joints feel better. You'll feel much better in a week or two, but it takes the anti biotics a little time to work. Just be sure to try and get plenty of rest, stay hydrated, and try to keep your stress low while you're healing.
Your future syphilis tests are always going to be positive after this, but the doctors can tell if you have an active infection or not based on some stuff in the test. If you're asked if you've had syphilis before next time you get tested let them know.