It's legal here in Arizona, but not in Nevada, Utah, or California (not sure about New Mexico) which are all states bordering Arizona, which might limit your travels.

However, I think if one has a road plated ATV from a foreign country, such as one of the European countries, you might get away with using one on the roads across the US. I remember a German guy rode from the east coast to California on a quad, and continuing across the world and the article was in the Los Angeles Times, but I don't remember the specifics or the time period for that matter.

Yes, the Trans Am Trail is about the best bet for a mostly off-road southeast to northwest route clear across most of the country, and there are numerous threads here and on advrider about the subject. I think the catch is that one needs to get a map for each state's part of the Trail as there is not one map that covers the whole thing (as far as I know).

If you wanted to regionalize, I know that there are certain states which would probably make a good off-road trek. Having lived in the northern part of Arizona on the Navajo Indian Reservation, I know that there are endless dirt roads/trails throughout that part of Arizona, and if one was to research, could connect with the southern part of this state in one big loop over a week's time.