"Me" has things done to/with them, so "it was me" is correct, because you'd say "they arrested me".
"I" us the one who does things. "I punched Rap in the nose, so they arrested me".

The reason it's "Bill and I played music" and not "Bill and me played music is that you'd say "I played music" not "me played music."

Of course, if you're in Jamaica, I think "they arrested I" is correct.

In the end, if you have a problem with someone else's grammar and usage that gets in the way of communicating, YOU have a problem, not they. ("They"--awkward, or what?)