Does anyone know if a registered sex offender would be allowed to live in the home with his own biological kid, if he became a father?

That would depend on what state the RSO lives in and on the exact nature of the offence for which registration was mandated.

I would say yes. Many registered sex offenders have children after their sentence or they can regain custody of the children they already have once released. The only time I don't think they can live with the child is if that child is the one that the offender molested.

It is important to remember that not everyone on the registry is a predator or molester.
The label "Sex offender" covers an awful lot of ground and not all of it is muddy.
I have a friend who is an RSO in California because he lost his head over this georgeous thing he met on his route- He was a beer delivery driver, she was tending bar... turned out she was 17, so he got nailed with a statutory rape charge, went to prison and must register as a sex offender even though he had no idea at all she was underage, and no reason at all to doubt her word that she was over 21.
The registry does NOT mean a person is horrible, it means you should take a closer look than you might have otherwise done.

That is very true. I know someone on the sex offender list for indecent exposure. He got caught having sex with his girlfriend in a public place. That put him on the list.

See here

Thanks a lot. Actually, the person I'm referring to shouldn't be a registered offender either. It's very complicated situation.