You are correct that in most nations single doses of amoxicillin are no longer recommended for treatment of gonorrhea.
The symptoms you have suggest that you have either gonorrhea or nongonococcal urethritis. Without a test one cannot tell which is present. If testing is not available, the recommended treatment would be either a single injection of ceftriaxone or cefixime in a single dose of 400 mg, FOLLOWED by either a single dose of azithromycin or 7 days of doxycycline 1000 mg twice daily. this therapy will provide reliable therapy for both gonorrhea and NGU. Chlamydia is not a concern as it does not regularly infect the throat and therefore cannot be acquired from oral sex but, if it was there, this therapy would cure it as well..
In answer to your specific questions:
1. In some settings if testing is not available, health care providers will just treat following observation of a discharge.
3. See above
Hope these comments help. EWH