Is the OTC male fertility test accurate?

Yes. Not a full test. The over-the-counter fertility test for men is called spermcheck and tests only sperm count (concentration): low or normal. Accuracy is ok. A full semen analysis (sa) through a fertility clinic looks at sperm count as well as motility (% that are moving) and morphology (% of normal-shaped sperm). Some clinics test sperm antibodies too. The home test is better than nothing but is not a full sa test.
Can be. They are quite different. Some detect a minimum number of sperm; some detect motile sperm. Here is a review of them: http://theturekclinic.Com/2012/02/13/male-fertility-kits-semen-analysis-sperm-counts/.