There are a lot of dogs that run loose and chase bikes in Missouri and Kentucky, but I didn't consider them a big problem.
People everywhere offered us food and cold water and were generally kind and generous.
As far as the 45-50 days... Some people do manage to do TA or NT that fast, but I think not too many.
The ST would seem like it might be a good bit quicker to finish, but plan to do it in cooler weather if you go that route.
As Valygrl says the Pacific Coast isn't coast to coast, but does cross the country north to south and is very nice and can easily be done in you time frame. You won't see the whole country that way though.