TLS certificate generation on https webserver?

Question

Web servers

Every time users connect to the https web server, does the web server generate a new certificate from the private key of the certificate? That is, each time a public RSA key is generated from a private one?

Answer 1

No.
The client downloads the certificate along with the public key to validate the certificate and encrypt the shared random secret to be used later.
A new RSA is not generated every time.