Mozilla and Chrome do, but disabled by default. Chrome recently tried to enable it and they had to notice that a bunch of crap-devices by bluecoat prevented it from working: https://bugs.chromium.org/p/chromium/issues/detail?id=694593

Yes, with the caveat that TLS 1.3 is not finished. So nobody (including nginx) has a completed TLS 1.3, but they all have drafts which are presumed to be close to the final thing.

Builds of at least Firefox and Chrome / Chromium with TLS 1.3 exist in the wild if you're interested in testing this out.

In particular you should look at this if you have middle boxes such as a "transparent" TLS proxy or "traffic inspection" feature in a corporate or educational network. It is very likely these boxes have defects regarding TLS 1.3 because all of them are garbage built by idiots. If you find out early you might be able to plan for what to do about that before the proverbial impact with a fan.