Katze is also regularly used of any cat, regardless of the cat's gender, if the gender is not relevant.

And traditionally, masculine titles for professions such as mein Chef were also used when the gender of the person was unknown or considered irrelevant.

Yes, you could use meine Chefin if you wanted to.

Regarding the grammar or the content? ;)

Because the word Katze is feminine, so you need to use diese Katze, not dieser Katze here.