Private key ECDSA/TLS 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.