Electrical – Tv cable, safe to cut

Cable installers work with "live" wires all the time. There's no voltage on their lines that I'm aware of, only signal and maybe some interference. You do need to disconnect lines for satellite dishes or mast mounted amplifiers since they may have a power injector that adds voltage into the lines.

When crimping the new cable, make sure the center wire is long enough (you can trim it after you crimp a new end) and get some of metal mesh that forms the second wire in tight contact with the crimped fastener. I usually pull some of that mesh back over the insulation so there's a tight bond when it's crimped.

I suppose it's safe as long as you wire it properly, including a secure attachment between the box and the cord to prevent the cord from pulling out. It certainly is not code-compliant, but temporary wiring like an extension cord isn't covered by electrical codes I believe. Yes, I would ground the box. (Depending on which outlet you install, it may be grounded automatically with the receptacle mounting screw, but I would do it with a pigtail anyway.

But why wouldn't you just use a power strip instead? Definitely easier and safer, probably similar in price to buying the parts individually.