They do not lose their parental rights by virtue of their status as a sex offender. Their parental rights can be taken away by court order with justification.
Not necessarily; as with nearly all things in the court system, the activities of the various branches do not automatically coordinate with judgments in other branches. Somebody needs to ask them to react, which here would involve the filing a separate lawsuit in juvenile court for termination of parental rights. TPR (termination of parental rights) cases are major, expensive litigations in which the defendant has a right to a trial by jury, so they are not undertaken lightly. However, if anyone does bother to file one, their odds of winning might be improved if the defendant has a sex offender record.
Do not assume that I am your attorney because of my response here, since I would not normally be taking any further action on your case. You can request further clarifications on or past answers which might help you at http://www.attorneyqna.com/attorneys/53401-wi-jay-nixon-1529181/answers.html?sort=recency http:// or at http://www.lawguru.com/answers/search/attorney/jknixon.
Parental rights need to be formally terminated in separate court proceedings. If that has not happened, there are two sources of restriction on a sex offender's contact with their own children. First, the direct and collateral consequences of the conviction, such as conditions of supervision and legal constraints placed on registered sex offenders by statute. Second, any restrictions imposed by the family court.
If you have already had your parental rights terminated, then being released from custody does not bring those rights back. If you have NOT had your parental rights terminated, but have a court order that does not allow for visitation, then you can petition the family court to change that order now that you are out of custody. However, just because you are now out of custody does not mean that the judge is required to change the order. While reunification with biological parents is often a goal in family court matters, it is not the only goal. The best interests of the child are the most important consideration and there may be any number of arguments why someone convicted of "raping" a 16-year-old might not be the best person to have around children. I would suggest getting an attorney who is very experienced in dealing with child custody issues. Good luck.