I'm Lucy, and I'd be happy to answer your questions today. I'm sorry to hear about your situation.
You unfortunately cannot adopt your daughter over the father's objection unless the mother gets a court order terminating parental rights. A stepparent adoption does automatically terminate the obligation to pay child support, so if the biological father won't agree, telling him that might be one way to get his consent. To terminate his rights over his objection, it's necessary to show that he's unwilling or unable to protect the child from harm, he's been unwilling to take responsibility for the child, or the child is abandoned. Maine Stat., Title 22, Section 4055. The fact that he's paying child support could count against you on those last two, because making payments is a type of responsibility and abandonment requires a total cessation of all contact. But the mother could try sitting down with a local attorney to review all the details of the case and discuss whether that might be the way to go.
The mother can apply for a name change without an adoption when it is in the best interests of the child to do so. One reason for arguing that it's in the child's best interests is if you are married to the mother and she already shares your last name - giving your daughter the same last name can help her feel like a member of the family. The father has to be notified and has a right to object, but if you and the mother are raising the girl and he isn't really part of her life, the law is on your side (especially because she can take the stand and say she wants to do this). Changing her last name will not affect child support because it doesn't terminate her legal relationship with the biological father.
This site has forms and instructions that might help, or a local attorney could help draw up the papers for you.
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