This so hard, I mentioned this in a post just last week. There's a 9/11 every month on US roads, yet we leave the number 1 safety improvement (computerized traffic) we could make to probably eradicate 90% to private business: pushing driverless cars. Now of course at the end of the day this change has to be driven by businesses, but government can and should help: R&D and prototyping (like a ton of big inventions, e.g. the internet) could and should be funded to a very large extent by the government. Commercialization should be entirely left to businesses, but even here governments can use regulation to introduce new safety standards to inform drivers of the safest options for driving at the very least. And you could even go so far as to employ a model akin to that of many governments which increase taxes on fuel inefficient cars (e.g. SUVs) and decrease them on fuel efficient cars as an environmental measure, but then apply the same on the basis of safety (driverless cars, if they score well on safety which they should over time, being cheaper). You can argue that such a move is big government, you can also argue that's simply the government forcing the market to price in externalities (purely from a financial perspective, road death and injury increasing costs of healthcare on society, decreasing productivity, but also things like congestion and fuel efficiency can be priced in, which driverless cars should do better in) that currently are ignored. It'd still be entirely market driven, but the market reflects reality better. Once it picks up individual states could choose to make certain sections of roads (e.g. one lane) or sections of their city, entirely driverless car only at certain hours of the day, to accelerate the process. And once you go driverless you'll essentially open up what 'public transportation' really means. In a sharing economy of vehicles which need no driver, anyone's car essentially becomes a tiny bus-service. I'd expect the cost of transportation to drop quite significantly.
Absolutely. I think they can do more. The Challenge was relatively (mostly because there wasn't much else) significant 10 years ago and had relatively small ($1m for the winner, per year) attached to them. Not quite the effort we'd expect in relation to the size of the automotive industry, or the damage of externalities of driving that could largely be solved or improved (loss of life, injury, congestion, fuel efficiency, cost of transportation etc). I'm also not aware they continued after 2007.
Of course I wouldn't be surprised in the least if all those university teams (e.g. Carnegie Mellon who came in first, or MIT who came in 4th) are either employed at e.g. Google or Apple working on these things right now, or provided or are providing the research to those who are.
edit: yeah Google's driverless car project is headed by the Stanford director whose team won the 2005 DARPA challenge.