It seems you can get TLS 1.3 (or atlest slighty modified 1.3) to be quantum secure, but it increases the handshake size by roughly 9x. Cloudflare unfortunately didn't mention much about the other downsides though.

https://blog.cloudflare.com/kemtls-post-quantum-tls-without-...

Yes-ish. They're not enabled yet, but post-quantum signatures & KEMs are available in some experimental versions of TLS. None are yet standardized, but I'd expect a final version well before QCs can actually break practical signatures or key exchanges.