Being a Yankee, I can comment with fair authority about all of the above, and other things as well. A paucity of knowledge in a subject never lets a good American stay quiet.
That said, you're talking about trying to cover a *lot* of territory in not much time. Just getting across New York will take a week and change, and then you've got about a 1,500 mile ride to get to anyplace civilized in Florida. If there is such a thing. Of course, by going to Florida, you can cross New York City off your list, because Florida is where New York goes to retire. Cycling-wise, it's just like NYC except the drivers are older and blind. Not that you can really tell the difference, to be honest.
Chicago has nice cycling and great blues. There's a couple of really nice jazz-and-ribs joints in Chicago, if you decide to visit there, let me know, and I'll clue you in. Chicago is billiard-table flat, btw, which is nice when you're from Edinburgh.
As far as getting around, forget the train. Trains are largely useless in the U.S., and they only go to places you don't want to be. They'll also give you a hard time about the bikes. Many Amtrak trains refuse to take bikes, but you'll never know for sure, because it's up to the conductor. Tell him it's a personal mobility device, threaten an ADA lawsuit and you might squeeze by, but don't count on it.
So to get from A to a distant B, either fly or rent a car and drive. And when looking at maps, remember, it's freakin' big. Put it this way: The state I grew up in, Ohio, is 14,000 square miles larger than the entire country of Scotland. And Ohio is only one of 50 states. You could fit 3 Scotlands in the state of Colorado,and still have room for a hamburger and fries.
In terms of familiarity, New England--particularly the mountains of New Hampshire -- will probably seem the most familiar to the Scottish visitor. If you can, head out west. See the Rockies, the Grand Canyon, the Black Hills and the Badlands. There ain't nothing like that in the old country, I can guarantee it. And the riding will be insanely fun.
Also, be aware that carrying concealed weapons (handguns) is perfectly legal and socially acceptable in many states, so pick your fights wisely, although in most cases, the cracker who pulls the gun on you will shoot himself in the ankle long before any harm befalls you.
Having said all that, please do come. You will find that Americans, for the most part, are loud, fun, outgoing, friendly and large. And a crapload of us are descended from Scots (Scots were actually the largest ethnic group to emigrate here until the vast wave of immigrants came up from the south. So now you have folks with names like Juan McDougall and Ian Ramirez. It's another one of the beauties of this country, it really is a huge melting pot).
Any questions, shoot me a PM. I'll answer anything, and I'll probably tell you if I'm making it up.