Taking into account all possibilities its too complex to be fully explained here, but here are my quick initial thoughts. (Note: I did not look up the laws):
I'm assuming no one would care that you made those modifications, so long as you didn't use the modifications on public streets and if there would be no chance of the system spontaneously taking over control (I was working on making my car similar to that but gave up).
I don't think there are many laws specifically targeting this at the moment, but I'm assuming it would be illegal to do currently, unless you had specific approval for testing (with huge restrictions).
Assuming it could be approved:
If someone was in the car to take over control at any time and be in constant focus of the vehicles travel, the chance for approval would be higher. Otherwise, unless the vehicle is fully autonomous (with restrictions), it wouldn't be allowed.
If it got in an accident, assuming there would be a person inside and, as stated previously, they maintained focus of the road, that person would be likely responsible. I don't think it is ever a good idea to blame the system (or person) driving the car externally, since that transfers the risk and liability to something that should ideally be more perfect than the driver themselves.
The problem with someone else controlling the vehicle is that they can make mistakes or have communication issues with the vehicle. Which makes that theoretically a bad idea.
Assuming its legal, which it would not be currently:
Who is responsible?
The driver inside probably (or if no driver, (in theory) obviously the owner/external driver)
Does this change if there is a person in the car who can take over
control at any time?
Yes. (see first answer) Even if the car is either theoretically legal or illegal to drive.
Does this change if there is a person in the car who can not drive?
Yes.(see first answer) But this probably wouldnt be allowed.
Does this change if someone else is injured or killed?
That would probably be the same as any other accident. Assuming what I already stated.
Does this change if there is a person in the car who can take over
control at any time and they are injured or killed?
If they could take over control I would assume they assume the risk, unless the external driver made a negligent or purposeful mistake.
Does this change if there is a person in the car who can not drive and
they are injured or killed?
Once again, IF this was allowed, it would be the external drivers fault.
What would happen if the car was pulled over by police and there was
no one inside, or if there was a person in the car who can take over
control at any time?
If this was allowed, there would likely be a way to contact the external driver, and they would be charged. Unless the driver focused on road and was able to take control.
