I’m an American citizen. Say I moved to the UK and lived there for some time. If I were to apply for British citizenship and successfully acquire it, would I retain my American citizenship?
Yes.
Yes. I have both. You used to have to choose (until the mid-1990s, I think) when you reached 21. This was a requirement imposed by the US. However, even prior to that point, ‘off-the-record’ advice would be just to keep both citizenships and not tell anyone about it.
I don’t think that the UK has ever had a problem with dual citizenship.