Yes, sharp here just means very strong and tangy. That's beautiful.
Yes, in the US, what you will find in most grocery stores will be mild, medium and sharp.
The State of Wisconsin dominates the US market for cheese, and they make good cheese -- the people of Wisconsin proudly own the nickname, "Cheesheads."
There are a few other States, namely New York and Vermont that are very proud of their Cheddars. English Cheddar is not hard to find here, either. It costs a bit more, but it had to travel a long way to get here.
As for American cheese, the cheap stuff labeled "cheese product" is not something I eat, now that I'm an adult, but it is what I grew up eating -- along with most kids in the US.
American cheese is the quintessential cheeseburger cheese in the US. I like it, but also like to mix it up a bit by using other cheeses, including Cheddar.
CD