To such a question like "What's this?", can I answer simply "A red pen" or should I always say "It's a red pen"?
It is correct to use a noun phrase "A red pen" in answer to such questions, except when a teacher specifically tells you to answer in "A full sentence". All the following are correct short responses.
Who is the British Prime Minister?
Which chocolate do you want?
The red one.
How do you play violin?
Why do you play violin?
When does the train leave?
At two sharp.