“Wailing” has been mostly replaced with “screaming”, which means pretty much the same thing. Also, it’s the instrument that is “wailing” or “screaming”, not the player (unless the instrument is a human voice).
In your presriptive opinion. In actual usage, it is almost universally “He was wailing on that guitar” or “he was wailing on that solo.” Nobody ever claims that the guitar was wailing on that solo.
Yes, it’s slang. That was the whole point of the Onion article. It’s a word used only in a certain community, in informal situations. But slang words still have etymology. Often, though, they have more than one spelling.