AIUI (Which is very limited understanding) the Pro-Bel they use in White City has a web interface which you can log in and move things about on the automation system. Don't hold me to that though.
It's absolutely possible, although I'd imagine that running a TV channel remotely would be a bit hairy and not something you'd want to do for anything important.
It's possible but has very limited use. If your control rooms are manned 24/7 then it will be faster to phone them to make a change than it will to boot up a laptop and VPN in. If it is an evacuation scenario, then whatever triggered the evac would have had to have left the facility fully functional - including the comms links to the datacentre where the VPN traffic comes in. It can happen, like the recent BH fire, but evacs are often triggered by things like power and network faults which will break the remote access too.
You probably wouldn't have any monitoring except what you can see off air... and that's too late especially if there's a significant satellite/digital delay.
The subtitlers used to (and may still) work from home on the smaller programmes, like the breakfast regional opts. They would phone/ISDN in to get the audio without the delay, then fall back to using DSAT if they couldn't dial in for whatever reason. This introduced a large delay, and could IIRC result in the end of the subtits being lost when the opt ends.