Would this sentence actually be used like this to compliment someone on a nice name?
It seems a bit weird, judgy to me; I wouldn't use it. I think 'Ich mag deinen Namen' (= 'I like your name') or 'Du hast einen schönen Namen' (= 'You have a lovely name.') would be a better compliment.
I thought it would be used in the context of, say, deciding on baby names. That name is good, that name is weird, and so on. The baby isn't named yet so it's not so judging.
To what extent would one use "Vorname" instead of just "Name"? In English I just speak of your "name", I'd hearly ever specifically say "first name"