An HTTP proxy won't proxy DNS lookups. If you're using SOCKS4a/5 proxying, Firefox will use it IF you set network.proxy.socks_remote_dns=true, on Chrome it'll work if you use socks5 but the dns prefetch will bypass it and expose you. On Firefox, plugins and extensions can bypass the proxy. I don't know specifically about Chrome, but I experimented with it and when playing around with strict firewall egress filtering and restrictive apparmor profile, using the embedded Flash would just up and crash the browser.

It's not a good idea. If you want to use the web on Tor, use the browser bundle.

And that's what the TOR browser bundle does. And that's why it is locked down and disables all plugins etc.

The point of hosting in docker in docker is to firewall the browser so it can't talk to anyone else, and to add extra layers of protection - however flimsy - against any eventual targeted compromise.

That wouldn't be good for your anonymity. You get a lot of fingerprinting issues with other browsers which the tor browser tries to avoid.