Is the private key required once the certificate is hashed and signed? After a certificate is generated, public key is shared within the certificate itself. Is the private key used for any other purpose within the handshake or certificate validation?
Is the private key required once the certificate is hashed and signed?
That of the CA that issued the certificate? No. That of the end entity you are trying to communicate with? Yes. The whole point of a certificate is to create a binding between an entity (eg a website) and a public key, so that by proving ownership of the associated private key somebody (the entity) can confirm to be the entity named in the certificate.
Is the private key used for any other purpose within the handshake or certificate validation?
Again, the CA's private key: No. The end entity's private key is used to sign a specific part of the handshake (usually the ephemeral ECDH public key) so that the party knows that said part really originated from the entity named in the certificate and that they are connecting to that entity which then also knows the negotiated symmetric keys.
Yes; it is commonly used to validate that the certificate is still valid. It may sign CRL's or OCSP responses. In principle other (trusted) private may be used as well, but often the private key of the CA itself will be used for this kind of purpose. The OCSP response can also be used by OCSP stapling.
Certificate validation is of course part of the handshake if certificates are used.
Note that there are other ways of validating that certificates are still valid such as simply assuming that a certificate is valid if it has been signed / pinned or otherwise trusted. It does not need to be used for anything else; for root certificates it is quite common to put the private key in a safe (the physical, unconnected kind, commonly made of large chunks of metal).