I’ve not read the whole thread, but I have had the same issues with my menu not working on twenty twelve – the issue in my case was a clash with Google Analytics (so my developer tells me:
“It was the Google Analytics that was breaking the iPhone menu. I’ve fixed the menu problem so go look at GA and make sure it’s still working there.”
Sorry I can’t tell you how to fix it, but I hope this is a good starting point for someone!
