The term comes from the 1800's in America. A horse is actually not a very intelligent animal. However, it is highly instinctive and follows a set of very common sense rules. It runs from danger, it tries to stay in a pack for protection, it seeks out food, it protects its young. These actions became known as "horse sense". So anyone with horse sense is displaying a good level of common sense.

It is common sense. Not that awesome - sorry

"See a man about a horse" ~ it means that you have to leave, but it wouldn't be polite to use the real reason. Usually nowadays it is "rarely" said when someone has to leave to use the bathroom.

I'm not sure what does mean exactly, but I think, that if somebody has a horse sense, that mean he always truly assess the situation.

NB: "slang" is not plural. ;)