If on Linux, you should look into what you can do with network namespaces. (I.e. exactly what you want, without the need for a container).

Sure. If you trust the results from the search engine your browser decides to use. Or in many users' cases, whatever search engine has latest hijacked your browser's search engine settings.

You should check out Defendor.

"Home" is pretty much Microsoft-speak for "Borked".