Congratulations on your upcoming baby.
There are really two separate parts to this question. First is whether your husband's name goes on the birth certificate. I'm sure your doctor was trying to be helpful, but placing a husband's name on the birth certificate is not necessarily an automatic procedure and there are ways to prevent his name from appearing on the birth certificate. The second question is how to get your boyfriend's name on the birth certificate. This can be accomplished...
There are very limited circumstances under which an individual's parental rights can be terminated in New Hampshire. One example of when parental rights could be terminated would be if a stepparent were willing to adopt the child. Please talk to a lawyer to find out exactly what the applicable laws are for your particular situation.
Congratulations on your new baby.
I encourage you to speak to an attorney because this situation can be tricky if both the child's father and DCYF are claiming that you are not fit to parent due to a medical condition. You can learn about some low and no cost legal services available to low income residents at: http://www.nhbar.org/for-the-public/free-legal-services.asp.
Congratulations on your new baby!
Please discuss this situation with a lawyer. You are smart to not want to do anything that a judge would frown upon, and a lawyer can help you come up with a plan of action that is right for your particular situation. There are issues here relative to the establishment of paternity, the establishment of parenting time and parental rights ("visitation" and "custody"), the the establishment of child support, and your ability to choose the child's last name,...
You can finds information on low cost and free legal services at http://www.nhbar.org/for-the-public/free-legal-services.asp and at http://www.nhbar.org/lawyer-referral/reducedfee.asp..
Good luck finding an attorney to help you through the process.
You typically need the consent of the legal father for a name change. Whether your husband or the man who contributed the sperm would be considered the legal father is a more complicated question and something you should discuss with an attorney.
For a myriad of reasons, it is not a good idea to communicate with a minor behind her parents' back in the manner that you propose. You need to remember that you are not her legal parent. I think the best thing for you to do would be to speak with either an attorney or social worker who is experienced with such issues so that you can come up with an appropriate plan to have contact with your daughter.
You should know that there are many circumstances where courts can recognize men as the legal fathers of children even when they did not contribute their genetic material to the child. However, these situations can get quite complicated, so I recommend you speak with a family law attorney right away to figure out the best way to proceed.