If SSL/TLS uses a pre-created certificate why does it also use public-key ciphers such as RSA and ECDSA for authentication (I do not mean MAC integrity)? Are they related?
The key in a TLS certificate is for a public-key cryptosystem. This is because the same key is sent to every client, and sending out a secret key to anyone who asks is not considered a viable security strategy by most experts.
The reason the certificate is signed with a public-key algorithm is that there's otherwise nothing stopping an attacker from pretending to be the real site, and giving you their public key. So, you want to make sure the public key is the correct one, which you do by verifying that it's been approved by someone you trust. This is the exact use-case for digital signatures, so they're used to verify the certificate.
The public key in an X.509 certificate is normally a public key for a signature algorithm, to sign data and (if the certificate is a CA certificate) to sign other certificates. In TLS, it can also be an encryption/key-exchange algorithm public key; the most common is an RSA public key, which can be used for both. In RSA-based cipher suites, this is used for key exchange. In DHE/ECDHE cipher suites, it's used in key exchange by verifying a temporary Diffie-Hellman key signed with the associated private key. Regardless, the end effect is the same: the public key in the server cert is for key exchange, and in CA certs is for certificate signing.