Not infrequently, the title of a new question contains all the information necessary for an accurate reply. That's the case here. Any and all existing HIV antibody tests, or combinations of antibody and viral tests (p24 antigen and DNA/RNA), are 100% reliable at 13 weeks. In fact, all available tests are accurate by 4-6 weeks and afterward. So without reading your actual question, I can reassure you that if you have had a negative test result 13 weeks after your last possible exposure to HIV, you are definitely in the clear.
Now I have read the question. Nothing has changed. To the specifics:
1) Yes, 100% conclusive. The HIV tests are among the most accurate, robust diagnostic tests ever developed for any medical condition.
2) There are no medical conditions that have any effect on HIV test reliability, with the possible exception of deadly or terminal problems like septicemia, terminal cancer, etc. Peptic ulcer disease or Helicobacter pylori have no effect.
3) Same for medications. There are none that affect HIV test reliability, except maybe for the kinds of drugs used for the medical problems just mentioned, such as potent chemotherapy or immune-suppressing drugs.
In case you are interested, here is the link to a recent thread that discusses why testing at 13 weeks remains officially recommended even though earlier results are conclusive:
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