> It even more strengthens my impression than I should first
> play with reconfiguring the kernel.

I have loaded the new Gentoo system using the kernel from the old one
with no change in Firefox behaviour on https://www.privat24.ua/#login
page: it crashed as was described before.

So, playing with the kernel configuration will not help. :(

I think this "This looks like a radeon video driver problem. You could go into a loop of rebuilding xorg, mesa, dev-libs/nss, what-ever and see if things improve" is the best suggestion so far.

I would do something like 'emerge -1 xorg-server xorg-drivers @x11-module-rebuild mesa llvm clang' then restart X and try again.