can I change the private key without changing the public key?
No. You can reuse the same private and public key for another certificate but you cannot switch only the private key and use the same public key. If this would be possible then pinning to a public key for security would also not make sense.
What would be the point of rotating the cert only to update the expiry date?
What is the point of ending a certificate in the first place? To have the ability to create a new key and maybe adapt to a better key algorithm if needed. And because the owner of a domain might change so the previous owner (who has the private key) should no longer be able to (mis)use a certificate for a domain it no longer owns.
Of course, this could be done with certificate revocation. But revocation does not scale since more and more revocation information would have to be stored and checked. An expiration thus provides a way to cleanup older revocations (since the certificate is expired anyway) and with a short expiration it also limits the impact if revocation is not properly checked in the first place - as it is unfortunately the case with most uses of TLS.
Thus, it is not necessary to create a new key with each certificate but it is good to have the option to create a new key if necessary.
... but then is it recommended to have a long expiry date on the next rotation?
It is better to have short expiration but change the key only when needed. And in case of public key pinning one should also have at least a second key which then gets used for the new certificate in case the first key was compromised.
When keys are rotated (say I leave the public and private key the same); is there anything mathematically or from an encryption perspective that changes if only the expiry date has changed?
I'm not sure what you mean. There are no changes to the keys but the certificate is newly signed by the issuer to proof that the existing information in it are still valid. Thus at least the expiration time and the signature on the certificate has been changed.