Pay the money to one of the background sites and find out why he spent time in jail.
Go to your attorney now, even if you take no action, just to let him know the circumstance and get his advice.
Not sure about this one, but if you have a custody agreement don't you have a court appointed person who is supposed to be on the childs' side?
On the one hand, no good parent tries to deprive their child of the other parent. Without good reason. Still the childs parent and they need both.
On the other hand, yes, according to statistics a big percentage of child abuse is commited by a step parent. So be on your guard.