1. They are a professor and a writer.

2. They are a professor and writer.
3. They are professors and writers.
4. He is a professor and writer.
5. He is a professor and a writer.
(Which expressions are not correct?)

3 answers

They are all correct. However, the exact sentence depends upon your meaning.
What about #5? Doesn't 'a professor and a writer' mean two persons?
No. The professor also may be a writer.