We met at Nick's house before going out to eat.

Proper Noun=Nick's
Common noun=house

Byron Elementay School has a day off tomorrow.

Proper noun= Bryon Elementary School
Common noun=tomorrow

They studied the pictures of Venus.

Proper Noun=Venus
Common Noun=They
Is this correct?

In #2 the common noun is day.... tomorrow tells when and is an adverb.

#1 is fine

#3 Venus is a proper noun
However, "they" is a pronoun.
So, what would the common noun be?

Yes, you are correct. "Venus" is a proper noun as it is the name of a specific planet, while "They" is a common noun as it refers to a general group of people. Well done!