I live in Wasilla Alaska.
I've lived with my husband of 10 years and current boyfriend, for about 5 years.
My youngest daughters birth certificate states that I am her mother and that my husband is her father (Alaska law states that a married woman who has a child, the husband's name is listed as the father on the birth certificate).
The child is actually BIOLOGICALLY the child of myself and my boyfriend.
There has NEVER been any legal action taken against anyone in regards to this detail OR the cohabitation.
In point of fact, I don't think I've EVER lived anywhere that we didn't have "cohabitation" of people who weren't married to one another in my whole adult life.......... in my whole life-I've lived here basically my whole life..
I would HIGHLY doubt you will have issues with ANY of that.
As for the babies name-ultimately-by law, only mama gets to choose that.
BUT if it were me, in your case I can't advise, 10 months is a short amount of time.
I gave my oldest child her bio-father's last name, 2 months after her birth he walked out-she's hated that name her whole life (she's 18).
I gave my other two children my husbands last name. They have no issue-but he's not going anywhere either, 11 years into this, they will always have their daddy...
"Love As Thou Wilt"