1) im not sure it has to be wired.. i cant think of any reason why you cannot connect to a wireless connection, and share that connection wireless as well.. we do it all the time with phones, tablets, and laptops.. granted i have not tried this out in real world, but unless it was specifically locked out, at a hardware or software level by the mfgr, i dont see a reason why it has to be wired. having said that, i think this would be a double NAT scenario though, so im sure you would have some routing issues to work out. but depending on the router (or router software) being used you might be able to add some static routes. definitely not for the feint of heart, you'll need to know what you're doing to get traffic flowing properly in some situations im sure.
2) power would definitely be an issue for the router, and possibly the car hotspot so that it works when the car is off. i realize not everyone has one, but my car has was built with a hotspot in it. i can poll my vehicle while off, to check tire pressure, oil life, lock/unlock doors, and remote start via the hotspot modem, again while off. so there are some hotspots that work while car is off, the extent of that i have not looked into though. i suspect that it is very limited feature wise while off. it might very well be easier to buy a third party hotspot to "hack" it into working than trying to get a ford/chevy/hyundai/toyota integrated hotspot working.
wifi eating up all the bandwidth, so none left for bitrate is definitely an issue. potentially not an issue for a different camera mfgr, but definitely an issue at the moment in this scenario for sure. i suspect that there is some bus sharing going on and that wifi is using up all the available bandwidth on the bus. thus leaving only a little left for encoding.
there are some issues to work out for sure. seems more trouble than it is worth at the moment!