Thinking about it, a wild hare makes more sense than a wild hair. Rabbits appear to act crazy during their springtime breeding season (which explains references to March hares and Bugs Bunnies) so acting like a wild hare would be a natural metaphor to rural people. Saying somebody had a wild hare up their ass would just be a more scatological version of that metaphor.
I’m three hours from Indiana and never heard it. But if I did, I’d turn to the speaker and ask “What does having a stray hair in your rectum have to do with dangerous or wild behavior?”
Unless it makes sense to Hoosiers to think “Gee, I feel like I’ve got an ingrown hair in my butt – I know! I’ll leave work in the middle of 3rd shift to go fishing 80 miles away, then drive an hour each way to get a bowl of spicy stew at the Mousetrap.”
The traditional expression is definitely hair and not hare.
The one that makes more sense is very often a correction. Consider “just desserts” or “another thing coming.”
I personally always thought it was “hare,” and even pulled up memories of seeing it written that way when I saw the question. The same for the other two.
I’ve heard “wild goose chase”, which I think some people are using “chasing a wild hare” to mean. But getting a “wild hair up his ass” is different. “Goose chase” means you are looking for something you’ll never find. “Wild hair” (which I’ve often seen as “getting a bug up your ass”) means to get the urge to do something weird or wild out of the blue. I’ve always used the “hair/bug” version, but I wouldn’t be surprised if another version came first and evolved into several slightly different versions.
I always knew it as “he got a wild hare up his ass”, and I really did always associate it with the animal, though I’ve no idea if that was just because I was little when I learned it or what. Also, if it matters, I grew up in rural Pennsylvania, up in the mountains in the Northeast.
I grew up in California, heard it fairly often, and always thought of it as “hair” but upon reflection it could easily have been meant as “hare.”
I also assume I heard it as often on broadcast radio or TV as IRL so I’m not sure regionalisms are particularly discernable.
In all fairness, a wild hare up there would make a person fidgety too.
I think you are on the right track-
I’ve always understood it as “wild hair”. Google ngrams shows “wild hair” much more common than “wild hare”, by a factor of 20 or more, but it’s not very meaningful because both phrases are commonly used in other senses than in this idiom.
There’s also a wild heir.
Wild geese fly long and far away. A wild goose hunt may take you away for a while.
A Snipe Hunt is hunting for a non-existing object, (like striped paint, or a long weight).
Coursing will have you Haring Off in all directions at high speed, going off at a tangent or even doubling back. Brown Hares can go 30-40 Mph
Things I learned today: the Jack Rabbit is a kind of Hare. Up to 45 Mph. I don’t know if JackRabbits are known for the abrupt turns like English/European Hares?