I can confirm that VP9, HEVC and H.264 encoding/decoding work when using the VA-API. Can't comment on AV1 hardware acceleration as my RDNA1 card does not support it but no reason why it should not work. Also keep in mind that in both fedora and opensuse, the official Mesa package does not contain the code needed for HEVC and H.264 encoding as these are not royalty free and hence have been removed in fear of a lawsuit. You can replace these Mesa installs with community packaged versions which have support for all codecs.
Arch, debian and other distros don't have this problem.
Edit : Regarding the browser support for video acceleration, Firefox and its derivatives, support can be enabled easily(but it's not enabled out of the box). Chromium supports it as well but I have not tried it as I use Firefox and have only dabbled with chromium