See: https://github.com/http2/http2-spec/wiki/Implementations. nghttp2 is solid.

Facebook's proxygen has HTTP/2 support "in progress": https://github.com/facebook/proxygen

Yes, but I believe they don't support websockets yet. At least, searching their github for "websockets" gives only two broken links.

UPDATE: I noticed somebody wrote websocket support [1], but it didn't get merged yet with the master.

[1] https://github.com/kekekeks/proxygen

If you need only realtime push then Server-Sent Events work over HTTP/2.