It's like international waters beyond 20NM.
No you can't shoot. Shooting is an act of war, just like sinking a ship in the mid atlantic.
Or, what would happen if I sent my own satellite, and it hit Elon's by accident?
I am asking these questions, because I am not sure big countries would be very happy if their people's internet requests went through USA's company.
Not entirely necessary. Most nation states have enough property spread over the world that suing them can be sufficient. For example: http://www.nydailynews.com/new-york/feds-win-seize-ave-tower...
But I don't see why they would wanna do that. I don't know how these things works but it seems like there's enough space in space for all the satellites SpaceX would wanna launch.
Russia does :) https://en.wikipedia.org/wiki/Dnepr-1
100km is the altitude where lies the Karman Line[0]; the boundary between the Earth's atmosphere and outer space. So technically, space IS what's above 100km ;)
Similar rules apply.