From the British end, I've never heard "do a fart" either. "Have a fart", at the most, but usually just "fart". Oh, and of course, "to let off a (XXX) fart" (useful if you want to put an adjective where the x's are)
I've never personally heard of "doing a fart" in the sense of "something going wrong" but I can believe it. In fact, that was exactly what I, as a child, thought was happening when our 1960's Volkswagen Beetle used to backfire.It depends a bit on the context, I think.
"You're making a funny face. Are you trying to do a fart?" is not a really natural phrasing, but I can imagine someone saying it that way.
"I was driving down the road and then my car did a fart. The transmission dropped right out of it." This meaning/usage could be applied to anything going wrong, whether or not there is a smell or a bad sound.
technically it's "flatulence" but I would say the majority aren't familiar with the term, so "pass gas" is commonly used or "break wind"When I was a child, my mother (like Pops' family) thought "fart" was too vulgar, so she always said "to pass gas," which I believe is the medical term and is still used whenever sensitivity is desired.
I love it when students come out of their shells like that, it makes it all worthwhile.The single best compliment of my academic career came for a very intelligent but almost painfully quiet young woman who, having never participated in discussions in several of my classes, told me after my presentation that "that was a fucking good lecture today."