I've never heard about not exceeding 5 miles on mountain tires. I know people who have ridden centuries on them. They are, however, not terribly efficient. You can find nice tires for your bike without knobbies, and that will remove that limitation, though they will likely cost more than your bike does...
On the traffic side, do some web searching to see if there are any resources, or ask at a bike shop. I also live in an area with a considerable amount of traffic, but there are routes you can ride that avoid most of it.
Now, on to the question about the stationary bike. Stationary bikes are okay from a fitness perspective, but most of them don't fit people very well, and they are - in my opinion - mind-numbingly boring. Having said that, with a book and some music, it may be tolerable, and it's not a bad way to get into better shape. Note that if the bike displays the number of calories burnt, it's generally wildly optimistic.