Yes. Learning English with materials or a teacher from the USA won't damage your understanding of English from other countries. ;) We do understand each other (well enough)!

Yes, most of the time, but sometimes when people have a very thick Cockney accent, I can't understand them when they talk fast, but if I listen for a few minutes I can then understand. In contrast some British accents sound almost American to me, very generic, like on the BBC. They pronounce words only slightly differently, but not so differently that we don't understand each other. In my mind they sound much more proper and sophisticated to me.

Yes. U.S. and British English are mutually understandable. A native U.S. speaker sometimes has minor difficulties understanding or being understood by British speakers, but a native U.S. speaker can have just as much trouble with an unfamiliar U.S. regional dialect. Joke saying: "A language is a dialect that has an army and a navy." "British" and "American" English are NOT very different. The only reason why they are considered separate is that the U.S. and the U.K. each "has an army and a navy." That is, there are separate (UNOFFICIAL!) national standards for the two languages. The Oxford English Dictionary series is the traditional standard for British English, the Merriam-Webster dictionary series is the traditional standard for American English. If the United States had not separated from England, we would probably still have American English. It would be just a dialect, with no special status. National pride created a marketing opportunity for a "Dictionary of the American Language." Lexicographer Noah Webster seized that opportunity. And now we have separate standards for American and British English.

Yes.