Used as a name, Jen is proper.
proper
As a name of a road , Park Avenue', it is a proper noun, and both words star with a capital letter. However, when used separately, as 'the park, or 'the avenue', they are common nouns and so not need a capital letter.
common
common and proper
It is a proper noun, because it is the name of a specific thing.
common
it's a common noun. a proper noun would be Spider-Man.
After "Jen" - Is it snowing Jen?
Jen Martinez goes by Jen.
Proper
Pencil proper or common noun
It's a common noun as it's a general name for a thing. There's nothing special or particular about it.