I'm really fascinated by this because some of the comments I've been reading here are very opinionated but I don't know what they are based on.
I doubt this was part of the curriculum, but in fitness classes at school we were told that breathing through your nose with your tongue pressed against the roof of your mouth helps develop your face/jaw and that breathing through your nose moistens the air and helps filter it. We were also told that it's not necessary to breath through your mouth when you're out of breath as you won't get or don't need more oxygen and that you risk hyperventilating this way, instead we were told that you should breath using your diaphragm to help mix the air in the lungs and that you should hold the breath for about as long as you inhale/exhale if you can help it. This sort of breathing exercise seems pretty common in any fitness/exercise related guide I've seen too.
As far as other countries/cultures go, any country that has a strong yoga/meditative culture likely considers breathing through your mouth to be bad too, but this is just speculation on my part. Pranayama is thousands of years old for instance and is primarily concerned with controlling and reducing your breathing, which is hard to do if you're not breathing through the nose.
My first exposure to the term was via internet memes where “mouth breathing” was conveyed as existing only with those who were mentally impaired. Check out the Urban Dictionary link someone else responded to you with in a different comment thread for more context.
I don’t recall ever hearing anything about “mouth breathing” in my society/culture/subculture before exposure to these memes, so I wonder if it’s a newer thing? If so, silly memes really are influencing our culture!
Yes and no. Mouth breathing in and of itself isn't, socially, a negative.
However, in English[0][1][2], a "mouth breather" is also a euphemism for someone of below average quality and intelligence.
Not sure how that came to be, but that's the answer to your question.
[0] https://en.wikipedia.org/wiki/Mouth_breathing
As for the current mouth vs nose and belly vs chest breathing, it really feels like a fad/cult. There seem to be a constant stream of very small "studies" with questionable methodologies with just enough results to warrant further investigation.
I always thought "mouth breather" was referring to stupid, but tough guys who did so much fighting that their smashed noses don't work anymore.
Also, my nose has been broken a few times and I can't ever really get enough air through it...