No AFAIK. The actual traffic is encrypted using a block cipher. Asymmetric crypto is only there to verify each peer's identity and establish a common secret for the symmetric cipher.
This is right - asymmetric crypto is awesome but just not anywhere near as fast as something like AES for chewing through chunks of data. So some anonymous asymmetric crypto (Diffie-Hellman or similar) is used to set up an encrypted channel, i.e set up a shared AES key, and then RSA is used to verify that the entity on the other end actually holds the private key for that subdomain. After that it’s AES pretty much all the way.
Yup that’s basically how it works. Usually when PFS is involved the keys are session based as opposed to using the same keypair for every session. This mitigates passive monitoring and future private key compromise/legal pressure permitting historic decryption of sniffed TLS sessions.