no
If it is a condition of your custody agreement, yes. If not, no.
No
Yes of course if he has custody.
See an attorney to file for custody. When your ex is found, you can use the court to force custody to you. If you already have legal custody, and your ex has taken the child in knowing violation of the order, then you should be bringing the police in, for non-custodial interference, or possibly kidnapping.
I believe your ex would collect it
I assume that the "ex-boyfriend" is also the father of the child. If you believe that he lied to gain custody, you can file a motion/petition for a custody re-hearing and be prepared to demonstrate that he lied, and that you are a 'fit' mother to have custody of the child.
file for custody
It depends on who has custody of the child.
If you mean, the child's mother (ex-wife) has custody, the child's father (ex-husband) is obligated or potentially obligated for support.
If your ex is the custodial parent then you are required to pay child support however if you have joint custody the order could change drastically. If you have your child as much as your ex and you dont have joint custody appointed by the court then you should file for it to reduce your payments.
For 1) are you in custody of your daughter? If you are in full custody you could take her to court for kidnapping is your EX mother in law in custody of your daughter? Do you have joint custody? If your ex mother in law has no custody over your daughter you could take her to court, she is not authorized to take care of your child.
you can turned that information into the internal revenue, and he or she will have to return that money, the only person should claim the child is the person that the child lived with through out that year.