Will public key change on renewing a certificate?

If you re-use the same certificate request, the key will not change. If you generate a new request, that'll typically (by default) generate a new key; re-using is sometimes useful e.g., to add another alternate domain name.

If your certificate authority is not requiring a certificate request, then they must be re-using the old one.

The certificate has nothing to do with the keys. Essentially the certificate is just a signature of your public key. You continue to reuse the same key (unless your old public key has an expiry date that's passed).