Simon, bicycling for transportation isn't as popular in the U.S. as it is in Europe, but it's not that uncommon. I make >90% of my trips by bicycle. If I were planning a trip likes yours I'd love to have a bike with me :)

Attitudes may vary wildly depending on where you travel. Some cities (e.g. Portland, San Francisco) are much more bike friendly than others (e.g. Atlanta, Dallas).

It is legal to ride on nearly all roads in the U.S. which are not restricted access (i.e. expressways), as well as many sidewalks. Having said that, it's not always wise to ride on many roads. Drivers can be downright hostile if they feel a bicycle is slowing them down for a considerable amount of time for no good reason.

Use good judgement. If traffic is light and cars can easily pass there should be no problem. If traffic is very heavy and the road only has one lane each direction, it might be best to take a different, less busy, route. And, most importantly, behave predictably.