If your posterior is exposed to the cold and you fart- will you be able to see your fart like you can see your breath?

Because it is much warmer than the outside air, yes. Hello Science Fair!

Absolutely, yes!

And now, from the lighter, yet true, side: Furthermore, if a lit match is held near the expulsion of those noxious gasses, they will ignite(!) just like the unburned fuel from the exhaust pipes of a race car. -I've seen it done, so I know it's true.