Have you tested positive for syphilis?
The info on secondary syphilis is pretty clear - https://www.cdc.gov/std/syphilis/Syphilis-Pocket-Guide-FINAL-508.pdf
The rash wouldn't start as a small spot. It may appear on the hands or feet first, but typically appears on the trunk or other parts of the body, too. The rash usually clears, with or without treatment, within 2-6 weeks. The rash usually appears at about 6 weeks, but the range is 3 weeks to 6 months.
Is there a reason you are taking doxy and didn't get the penicillin shots? Are you self-treating? If you haven't been diagnosed with syphilis, please don't give yourself antibiotics. For one, it will make it harder to test if you want to, and this is how antibiotic resistance happens.