Could syphilis cause hair loss and muscle ache and itchy skin and produce rash?

Yes: Here's your best approach. Lab testing for syphilis is reliable; get tested and if you need to be treated, accept treatment, which is usually easy. If the hair loss, aches, and rash persist, get with a good dermatologist / rheumatologist and see whether you perhaps have discoid lupus, vasculitis syndromes, and/or one of the other "connective tissue diseases" also. Keep up you proactive approach.