Can a sex offender at the highest level, sex offender being 17yrs old and the child was 5 yrs old when the incident took place, now sex offender is 32yrs old and still on probation, gain custody of his child..

He's a convicted sex offender. Unlikely that he'll get custody of the child.

What is the mother of the child like?

Also, where is this?

TheMotherHasNoCriminalRecordTX

MothetWantsNothingToDoWithHim

Is your space bar broken?

No, its unlikely he can get custody. Its possible, that he might get supervised visitation. Did the mother know of his record when she chose to have sex with him?

This is worded in a confusing and disturbing way. I can't tell if the victim is the father's child or another one.
'the child' and 'his child'
The child is now 20.
How old is his child?

At most, he may get supervised visits, with his child. I see no way he will get custody of his child.