Yes, they can drive on it, and yes, they can fly over it, if it's international. It causes serious problems for the Americans if the Northwest Passage is internationalized.
On the other hand, the Americans are far more concerned at this point with other international straits elsewhere in the world where there isn't a problem with ice a good part of the year. So far, they haven't managed to focus on the differences between the Northwest Passage and other warm water straits.
From our point of view, obviously we want to see it as Canadian territory and sovereign. But realistically, we need to face the international law fact that we might not win such a case, and we may be better off—may be—thinking of ways to have an internationalized joint responsibility in the northwest than trying to have it only for ourselves.
We also need to recognize the reality that if global warming continues at its present rate, in 10 to 15 or 20 years the Northwest Passage may very well be open water, and in fact the North Pole may be open water. If that happens, the Northwest Passage will disappear as a significant route. Why go through Canadian waters, through a torturous, winding course, if you can take an over-the-pole route that is shorter and doesn't get you involved with Canadian environmental regulations? That is probably coming in the near future.