VLC re encodes content on the fly and sends it to the chromecast, even the android app supports it (albeit making your phone hot). Not sure about your two requirements though, maybe so. else has a tip here.
Casting a browser or whole screen can be done with Chrome/Chromium on Linux. Whole screen doesn't work with wayland. Redirecting system audio doesn't work either.
If you redirect a tab which is mostly a single video file, it is smart enough to do the video decoding directly on the chromecast, so you get much better quality/framerate.
I was interested to find out that the youtube app on an amazon fire stick is supported as a chromecast target now.
It doesn't support casting a browser tab or a desktop though (the latter isn't surprising as I'm under wayland which the parent says doesn't work, but I'd guess a browser tab should work? would be interested in other peoples experiences of this).
Usually I'm interested in either casting a browser window, or even the whole screen usually, which I've been coming up short on.