Learn English – Let’s get the party started =? Let’s start the party

Yes, we do. You're correct that in British English have got is what's used, but at least in American English, we do say have gotten.

I've gotten 10 parking tickets already this year!

I have gotten almost no sleep since my baby was born.

In American English we use have gotten both when writing and when speaking. As Peter Shor notes, it sounds very odd to us to hear Brits say (or write) have got!

You might also be interested in this question on EL&U which asks about the difference between the two constructions and has some interesting answers.

Both questions are grammatically correct, but they have small differences in meaning.

In general, get + past participle is a way of expressing the passive voice. It is nearly identical in meaning to be + past participle.

Thus, "How did World War II get started?" has the same meaning as "How was World War II started?" This question asks about what forces or events caused World War II to begin.

"How did World War II start?" is similar in meaning, but the answers to this question could center around the events that occurred at the beginning of the war, as well as the events leading up to it. Either answer would probably be acceptable without further context.

As a side note, both of these questions are very, very broad, and the listener is likely to ask a follow-up question to narrow down the topic. A more focused question might be, "What were the major events that led to the start of World War II?"