Welcome to the forum. Thanks for your question. You had basically accurate replies on the community forum, but I'm happy to expand on the issues you raise.
To answer the overriding question, as posed in the title of your question: depending on the specific test done, or combination of tests, HIV testing is definitive at 4-8 weeks after the last possible exposure. Here is a link to a thread that discusses this in more detail, includng the reasons why official advice often remains to test at 3 months -- as usually recommended by frequent participants on MedHelp's HIV prevention community forum -- even though the scientific evidence indicates conclusive, reliable results at earlier times:
http://www.medhelp.org/posts/show/1704700
To your specific issues: This was a zero risk exposure for all practical purposes. Your HIV positive partner is exactly right: being on HIV therapy with an undetectable viral load makes him non-infectious. If there is any risk at all, it is extremely low.
As for the tests you have had, all are antibody tests, i.e. "3rd generation". At the times you had them, the results are 95+% reliable, probably closer to 99% for the tests at 36 and 37 days. And once you have tested negative on your viral load test (and you can be sure it WILL be negative), that combination -- a negative test for the virus itself plus the negative antibody tests -- that combination of results will be 100% conclusive (as discussed in the thread linked above). To your specific questions:
1,4) Answered above.
2) To my knowledge, in the US there is no approved rapid combo test. But you don't need it. The combination of your RNA/DNA test ("viral load") plus antibody is the equivalent of the combo test -- in fact, even better.
3) All your tests were 3rd generation. Only the combo tests are 4th gen -- in fact, that's the definition of 4th generation.
So no worries at all. Had I somehow been in your situation, I would not have felt a need to be tested at all -- your risk was that low. When your curently pending viral load test returns and is negative, your results will be conclusive and you can stop worrying.
Best wishes-- HHH, MD