Name changes and adoption for minor questions?

Say a woman does not get married and conceives a child with a man. Can that child have the father’s last name, or will it have to legally bare it’s mother’s last name?
Can a minor be adopted with having it’s last name changed to it’s parents/guardians?
How common is single parent adoption? (As in a single non-married person adopting a kid.)
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I’m writing a story and it’s recently dawned on me that a character’s last name may become an issue.
Bad typos: I meant without having to change it’s last name to accomidate it’s gaurdians/parents.
Not that the current answers are bad, in fact they helped a lot, but I wanted to add a little bit to help out the last question.
He’s adopted in the state of Maryland at the age of 14. I’m not sure if this helps, but I thought it wouldn’t hurt to add.