What is the level of the issur on mixed dancing? Minhag? D'Rabanan, D'oraita, etc.?
What about separate dancing but without a mehitza?
According to R' Gil Student, touching affectionately during a slow dance would be forbidden biblically (according to the Rambam) or rabbinicaly (according to the Ramban).
He says that there is also a ban on mixed dancing (even without touching) going back to the Maharam miRottenburg, and he says that R' Yehuda Henkin says that this ban is still in place he says that the fact that there is a ban is no proof that it would have been permitted without one, as many bans from the Rishonim's times were bans which re-banned already existing prohibitions).
He also mentions a Sdei Chemed that one is not even allowed to be present where women are dancing.
Dancing with one's spouse in public when they could touch each other in private: ill-advised. Shulchan Aruch warns against excess physical affection with one's wife in public. (In "Fiddler on the Roof", Tevye demands that the rabbi rule whether it is absolutely forbidden, and the rabbi hesitates; it's a "shouldn't." Some measure of privacy is called for in a healthy marriage.)
Or to quote Rabbi Elchanan Wasserman (as heard from Rabbi Rakeffet):
The Chumash ends with l'eini kol yisrael. Something can be permitted, but if done in front of the whole Jewish people, that's the end of the Torah!
Dancing with one's spouse at a time in the month when they're prohibited from touching each other -- prohibited. If the touch is "affectionate", it would be lo tikrevu legalot erva -- "don't even come close to prohibited relations"; the Rishonim debate whether that's a Biblical prohibition or a rabbinic one hinted by the verse. As for what's called "affectionate" touch ... well it depends what's called affectionate, and what's called dancing, I guess. (See Rabbi Henkin's essay for one Acharon who prohibited mixed dancing "as it may lead to affectionate touch.")
Having a big dance event for married couples -- ill-advised; on a given day, let's say a third of your couples are prohibited from dancing. Either they sit it out and it's giving away far too much private information (not to mention making them feel awful), or they feel pressured into doing something prohibited.
Dancing with someone else's spouse, or any single woman (who is presumed to be a nida) other than an immediate relative -- prohibited as above.
The mechitza -- serves two functions. One is to remind people to keep the dancing separate. (Again, just a matter of good policy, to the best of my knowledge; see Rabbi Henkin's essay for more discussion on when a mechitza is needed.) Also a matter of good policy; avoids "creepy guys staring at the women as they're dancing." If you're asking if an opaque mechitza is absolutely necessary because it's prohibited to watch the other side, well, there's watching and then there's watching.
It is forbidden because it is not richuk nashim but rather kiruv nashim. Source: "צריך אדם להתרחק מן הנשים מאד מאד." (from שולחן ערוך אבן העזר 21:1)
There is no issur against a married couple dancing together. And you will find in the Talmud that on Yom Kippur and TuB'Av, men would watch unmarried women dance to pick for themselves a bride. There is also no explicit issur against mixed dancing or activity in the Talmud at all, save in reference to the Beit Hamikdash. It therefore can not be a D'Rabanan or a D'Oreita.
Of course given the 'dangers' you will find a full gamut of opinions ranging from minhag to this is Asur of the worst kind from the Torah. However those who say it is Asur D'oraityah bring side off proofs like "listening to "the Rabbis"" (though many rabbis allow it), or "Lifnei Ever" and never bring any source for the actual prohibition of mixed dancing. It should also be recognized that if you aren't allowed to touch them you can't dance with them.
There is a story of the Bosterner Rebbe:
I was present at the wedding of one of the list members more than 20 years ago. Near the end, the bride and bridgegroom were dancing holding a handkerchief, and the Bostoner Rebbe (who was m'sader kiddushin) came in and removed the handkerchief, so that they could dance together.
R. Enkin specifically allows it for married people.
In 1960 in England it was common for men and women to dance together. Rabbis from outside the community wished to impose their view on the topic and it created a big scandal. You can see the documentation here. Local Rabbis, and Rabbis who knew the community allowed the mixed dancing even for singles.