Highly unlikely. A site could try to exploit unpatched security holes in your browser, but if your browser is up to date, this is unlikely to succeed. Modern browsers are very complex and large so they have lots of weaknesses, but they also get fixed quickly, a lot of eyes are on their code and they utilize sandboxing techniques as well to isolate things from your system.
Still, it's a good idea to harden your browser further yourself, or run it in an additional sandbox.
Check Flatpaks as well.
There will not be the equivalent compatibility at first. But there will be enough compatibility for most users to not care about it anymore. The equivalent compatibility will only exist once literally every hardware and software manufacturer supports Linux on their own as first-class citizen. And they will only do that once Linux has signifikant desktop marketshare. But it doesn't matter as long as most stuff still runs no matter what. Which is currently the case. And it's also gotten easy.