Did your testing include syphilis? Double check your results to make sure it did. It would have been a blood test.
To be 100% conclusive, it should be 12 weeks, but most people test positive by 6 weeks, and if you haven't had any symptoms, I'd be fine calling a 10 week test conclusive.
Loads and loads of things cause rashes. Syphilis is uncommon in most developed countries, and it wouldn't be near the top of the list for causes.