Yes. However "to be in hot water" is an idiom, it's informal, and it means a specific kind of trouble. It means "to be in danger of punishment." If I'm taking an exam in history, and I do all my studying about the 1900s, and then I discover that the exam is about the nineteenth century, then I'm in trouble, but I'm not "in hot water." If an exam proctor catches me looking at another student's paper, then I'm "in hot water." It is not used about a really serious situation. If a person is convicted of murder and sentenced to death, nobody would call that "in hot water," it would sound like joking about a serious matter.

Yes. But “to be in hot water” is more informal.

Just to add - it's not just AmE.

Yes

yes, they mean the same thing.