So do these batteries still suffer from the same problem of permanent damage if completely discharged? That's the biggest thing that scares me away from buying a Tesla (well, that and the high price of Tesla Motors models).
It may be a theoretical problem, but there are very sophisticated mechanisms in place to prevent this from actually happening in practice with current li-ion batteries. There was a big scare about it with Tesla vehicles a while back that turned out to mostly be one blogger spreading unsubstantiated claims.
It's true that if you keep driving an electric car until the car stops running, and then leave it for weeks or months without recharging as the battery pack continues to slowly trickle out, you could cause damage. Then again, if you keep driving a gasoline car for too long without changing the oil, you can also cause damage.
Not to the engine, but to the (low pressure) fuel pump. And that's because that pump is cooled/lubricated by the fuel. A few seconds won't hurt but longer is definitely not good.
You'll likely also have to bleed the system because the injectors won't open on air pressure alone, but that's not really damage (it just makes it a lot harder to get it started again once it is out of fuel).
