No, a biscuit in the US almost universally narrowly refers to a flaky, buttery, savory pastry typically eaten at breakfast, either with a sweet jam or a savory sausage gravy.

“Biscuit” is not a category of foods in the US, while “cookie” is absolutely a category of desert with many different kinds.

If you ask for a biscuit here, you’ll get the same thing every time. If you ask for a cookie, you’ll be asked “what kind?”

No. A biscuit is something entirely different in the US. We just have many types of cookies.