Are accurate male fertility tests available over the counter?

None I know of. The only real fertility test for men is a semen analysis which is a detailed look at the sperm and other cells in semen. It grades the sperm quality based on number, activity and shape. If the semen analysis is normal, there is no further testing necessary. When it is abnormal, a physical exam and some hormone tests are the next step.
Maybe. There are otc sperm tests, but using them to predict fertility is problematic. Fertell, kokopelli, and fertilmarq are all available. There is limited data correlating fertell to computerized semen analysis. We need more information on fertility outcomes.
Pretty good. Good but not great. Always good to see a urologist and to get a formal semen analysis to confirm what was found on the home semen test.