As for the bathroom debate, go to Europe. Public restrooms are unisex with lockable doors. No reason we can't do that here more and defang the debate, which has reasonable concerns on both sides
Far from a universal thing; common in small places like restaurants or wee cafes where you have no space so having two separate toilet rooms is the best use of space and at that point why bother gendering them (although lots of places still do, it baffles me) but lots of public conveniences are gendered and big multi-cubicle rooms.
They seemed pretty universal in bigger public accommodations in Sweden (department stores, train station, etc)
Sweden =/= Europe though; Europe is a very big diverse place and things are different wherever you go. I've never seen what you've seen in the places I've been (the UK, Belgium, Strasbourg, Berlin, Krakow and Lithuania - clarifying cities when I've only ever been to a single place in a country).
Yeah, been to over 20 countries in Europe by now (I think) and this isn't common at all.