How does the communication between the USB key and Google in a browser work? Will it work on all operating systems and browsers?
As for the browsers: Chrome/Chromium works fine. Firefox has an addon that adds security key support[0]. Unfortunately, this addon has a bug which causes high CPU load[1]. Firefox is also working on adding the native support for this, and I think that their ETA is somewhere in the second half of this year to get this thing to work natively. Unfortunately, even with the U2F addon on Firefox, you still won't be able to use it for Gmail, since Google hardcoded its browser to be the only one with U2F support, so you're going to have to change your user agent in order to use it, or default back to your second method of 2FA (Google Authenticator or something of the sorts). I have no knowledge about the U2F status on Safari and Edge.
As far as I was able to discover, the only service that didn't hardcode Chrome as being the only browser at the moment with U2F support is GitHub, and using YubiKeys with Firefox + U2F addon on GitHub works without any issues (other than the occasional high CPU load that I've already mentioned).
I wrote a short article on my blog about Yubikey usefulness for my usual setup about a year ago[2]. Things changed slightly to the better since then, but not by a lot.
[0] https://addons.mozilla.org/en-US/firefox/addon/u2f-support-a...
If you have a phone with USB Type-C and a Yubikey 4c you can just plug it in direct.
WebAuthentication will be supported in all major modern browsers [1], it just takes some time to implement.
[0]: https://bugzilla.mozilla.org/show_bug.cgi?id=1065729#c254