It means you have lost your right to physical and legal custody but you are still eligible to request visitation rights. Without parental rights you have no rights whatsoever in regards to your child.
Yes, when you give up parental rights that means all. Not necessarily - you might retain visitation rights. And you definitely retain the right to pay child support.
Unless you have the courts and the other parent's consent you will charged with kidnapping and the children will be brought back and you will lose custody and possibly your parental rights.
No. Custodial or visitations issues and child support are completely different matters. Parental rights can only be relinquished voluntarily by the parent or permanently terminated by the court.
That would depend on what the court thought of your relationship with the child.
No. I you have rights use them, that's what they are here for.
Don't do anything that will cause you to lose the right.
Difficult, but yes. In most cases, the state will take the child as single fathers have no assumed parental rights. see link
If your parental rights are taken away from you, you lose all contact and say over what that child does. You will not have visitation with the child at all and you cannot make decisions about their school, medical, or religion.
Not without permission from the court with agreement from the joint custodial parent.
When married you have equal rights to the child.
In most states you will lose you parental rights and if you want it back generally you will need a lawyer.
Technically the answer is yes. However, it isn't automatic; there would need to be a hearing, at which the apparent abandonment would be considered as a factor in deciding whether the mother should retain any kind of parental rights or not.