Well, I didn't find anything on wwh's link that hit the nail on the head, urinating or otherwise, for the verb "to minge." There wasn't anything on a plausible, but apparently non-existent verb "to ming."

There was an obsolate verb "to minge" meaning to mingle or to mix. I take this to be a social meaning.

Could this social mingling be the minging that you're talking about, not that the urinary application isn't also plausible?

However, we speak of a person who's had too much to drink as being "pissed," so wwh's original application of "minging" way up at the top of the thread still makes the drunken/pissed/urination connections all come together.

But I like best the obsolete minging for social mingling the best.