Do I need to file an extension if my state tax return still shows pending on April 15th and I’ve already paid what is owed?

No, you do not need to file an extension.  Your tax return was timely filed and the date you selected to pay the taxes owed was on or before April 15, 2024 so there is no issue.

No you don't need an extension. It was filed on time even if it is still "pending"--and you paid on time.  

No, if it is pending it is considered filed.  If it happens to be rejected, you then have a couple of day to correct it depending on which state you live in the time would be based on that state.