The NTSB could certainly do it. But they'd need to expand a lot in a world where self driving cars are an everyday reality, which again comes back to the question of funding.
It's a pretty bad look to have the involved parties directly fund the investigation. Some sort of general automated-vehicle tax may be more appropriate.
Not really. The EPA has done this for decades, specifically for Superfund sites. When you can, it's more efficient to have bad actors pay for their bad acts rather than burdening an industry as a whole.