Rather than getting confused, let me post an answer:
In both British and American English, the word "ass" is used for "donkey".
For "buttocks", British English uses "arse", while American English uses "ass".
In British English, the two words are not interchangeable. "Arse" means only "buttocks", while "ass" means only "donkey".
In American English, there is only one word, "ass", so the question of interchangeability does not arise.
The idiom about "making an ass of oneself" is present in both varieties and refers to the "donkey" sense. (Also, "don't be an ass", etc.) Unrelated to this, there is "arse about" in British English, but no "ass about" in American English.
Usage: The word "ass" is in practice less acceptable in American English than in British English, even when referring to donkeys, because of its other sense, considered vulgar.
(Aside: A similar but different situation occurs with "cock", sometimes considered vulgar in American English which prefers "rooster" for one of its senses. The British phrase "cock up" is not used in American English.)
