Mating behavior even in fixed male

We have a young male stray who we recently took to the vet, got fixed, got all his shots, etc. He's a very sweet cat, except he continually attempts to mate with our (also fixed) female cat, and he frequently bites without provocation (he doesn't actually seem upset or angry, he just...likes to bite). He seems like he might be 3 or 4 years old, so he's been a tomcat for quite a while, and I'm wondering how much effect fixing him will actually have in terms of these behaviors. It doesn't seem to have curbed them at all. He also continues to spray.

I'd really like to make him an indoor cat if possible, as there are lots of busy roads around here and I'm afraid of him getting run over. But the constant spraying and biting has got to stop, and he's clearly been an outdoor cat all his life.

Suggestions? I know that most people are in favor of making pet cats indoor kitties, but I'm not sure our new kitty would be happy that way. He seems very antsy/aggressive towards both our other cats and us when he's kept inside for too long.

If he's just been done, it takes about a month to get all the hormones out of his system. With it being spring, he still is thinking "tom cat". I would discourage the actions when you catch him.

Also no reason he cannot live inside. Many strays that have been outside cats are ok being totally inside.