Christian name = forename = first name = given name = font name = baptismal name. At least in American English these words are synonyms, though font name and baptismal name are from a religious context. Some people of some denominations might not use them or even understand them. First name is the most common usage in America.
What's the German equivalent to "surname"?
To my native UK tongue "forename" and "surname" are much more pleasing to say than "first name" and "last name" - Perhaps you North Americans should adopt/reclaim the terms? Then you could imagine you're really from Westeros or something... ;-)
When should I add the -n ending to Vorname since it is a weak noun?
Everywhere except in the nominative singular.
So: genitive singular, dative singular, accusative singular; nominative plural, genitive plural, dative plural, accusative plural.
(In the genitive singular, you add not only -en but also the -s that is common in masculine genitive singular: des Vornamens. Not all weak nouns add the -s in addition to the -en, but this one does.)
Does it not apply in this case?
No, because we are using the nominative singular here -- it's the subject of the verb and thus in the nominative case.