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.