Nah, school dances aren't common in Japan. Obviously I can't speak for everywhere, but some countries have those, others don't.
Anything that happens commonly in Japanese schools gets exploited over and over in school animes, so if they did have those, then they'd get exploited, but they don't. e.g. typical graduation episode in an anime is the boring diploma-receiving ceremony, and afterwards, the girls take the buttons from the uniforms of the boys they liked, while they stand around outside the school gates. This is a very common scene. If there was a "prom" then they'd show prom and not the boring diploma ceremonies. It would just make better television.
e.g. look at any American teen thing, and "who are you taking to the prom" is a very common plot-line. Basically, every American show ever with at least 1 teenage character has a "prom" episode, because it's a common experience that any show can tap into, and it's something that can be tapped for a lot of emotions. I've see a whole lot of school animes, and can't recall a single time that plot-line appears. Now, compare that to "who are you giving valentine's day chocolates?" or "who are you taking to the summer fireworks festival?". Both of those seem to happen in almost every anime where they could conceivably happen. If they did commonly have school dances in Japan to which people needed to bring a date, then animes would exploit that the same way they do the other stuff, but they just don't.