Can a gay couple, in which neither person has had sex, catch HIV if they are both tested negative but start having sex?

Unlikely. Not unless one of the couple has recently been exposed though non-sexual body fluids such as a needle stick or shared needles with IV drug use. If testing has occurred in the window period of HIV exposure, it is possible the test can be a false negative.
No. Hiv is a virus that can be transmitted only through semen, vaginal secretions, breast milk, and blood. In order for transmission to occur between two individuals, one must already have the virus. If both partners test negative and have not had any previous partners, there is no chance of getting infected.
No. One of the two men would have to already be HIV positive if transmission is to occur.