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.
In the UK forename is from the same source as before and is simply your first name. You will find now find either term on forms,etc. In the past forms/documents would normally say 'Christian name' but this ceased (after decades), to account for people of other faiths, or none. To be honest, over decades, 'Christian name' had effectively acquired a second general meaning (first/forename) unrelated to faith. At present, 'Christian name' is not uncommon in conversation, if appropriate.
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... ;-)
It can also be forename and family name, as it is in many other cultures.
Edit - I did not make this clear.
Forename is used for first name or Christian name (many people are not Christian so do not use this term). Family Name is used for last name or Surname in many cultures.
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.
When does one use Vorname and vornamen?
vornamen with a small v: never.
Vorname: in the nominative singular.
Vornamen: in the dative and accusative singular and in all cases in the plural.
Vornamens: in the genitive singular
Don't worry Lin, you are right and Duo is wrong. Not very helpful to use a programme that is actually quite often wrong even though mostly right. It is annoying and I have certainly made comments in the past. I too find Duo's non-thinking approach frustrating. Just know that you are right on this occasion :-) It is a bit like many of the 'English' spellings are not English at all, they are American. You just have to roll with the punch so you don't get hurt.