Should a pronoun agree in both number and gender with its antecedent?

Yes, it should. A pronoun must also agree in "case" with its antecedent. The possessive case of nouns cannot be antecedents for non-possessive pronouns.

An example of a faulty antecedent:

"The professor's desk was cluttered and he was trying to grade some papers."

It could be rephrased as:

"The professor had a cluttered desk, and he was trying to grade some papers."