Google doesn’t give me a satisfactory answer. Most sites that deal with the history of either the Symphony or the Minuet, merely mention that classical Symphonies included a Minuet, but they don’t provide a reason. Some smug ones offer “because aristocracy (haw haw, degenerate aristocrats)” as an explanation, but that doesn’t satisfy me as a real answer. Why not an Allemande, or a Gigue, or any other of the dozens of different dances that were popular during the 1700s?
Why a dance at all? “Because the Symphony/Sonata is an evolution of the Suite”, some sources claim, but I’m not buying that either, because early Symphonies didn’t feature a Minuet at all; they were in 3 movements.
From what I could gather, it was Johann Stamitz who first came up with the idea of adding a Minuet in his Symphonies. But why? Why a dance, rather than, say, a March, an Interludium, an Aria, or a solo improvisation? Did people actually dance to the Minuet? I know that Symphonies were performed at social gatherings, rather than in concert, so that might be a possibility. Are there any contemporary accounts that describe what people actually were doing during the performance of a Symphony? I’ve heard there was eating, talking and playing cards involved. But actual dancing?
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Interesting question. Here is another - why do classical minuets include Trios? That would seem to argue against the hypothesis that dance was involved, since the Trio would screw up the dance real good.
You could also ask why there were such a limited number of formats for the first movement.
I’d guess it was for the same reason sonnets have 14 lines. The form (at the time) called for it.
I wouldn’t be a bit surprised if they did actually dance during the “minuet” movement. Sort of an intermission, to break up having to sit through the rest of the symphony. Why else would they have included a dance movement rather than just a different tempo? I’d love to hear a definitive answer to this. Robert Greenberg may have mentioned this in his lectures.
I don’t know that there is a completely factual answer to your questions. Here’s what I learned in music school a lifetime ago while studying musicology.
The classic movement structure of a symphony is a fast opening movement, a slow second movement, a minuet and then a fast closing movement. The minuet provides a contrast of meter and tempo with the others. So, it seemed to hit the sweet spot in creating contrast with the other movements. It also provides - not exactly comic relief - but a bit of a break from the more serious movements.
Why a dance? Because there was a tradition of instrumental music being based on dance forms, even though they became so stylized that their direct connection with dancing was severed. Bach’s Violin and Cello sonatas are based on dance movements, but I think you’d be hard pressed to boogie down to them.
The quick google I did credits Haydn with the innovation. Given the esteem he was held in at the time, there is a reasonable chance it became widespread partially because he did it. Now, he also was a friend of Mozart and a teacher of Beethoven. So it could be that in actuality the symphonies of Haydn, Mozart and Beethoven have minuets (although Beethoven switched over to the Scherzo), and we are conflating that with the idea that ALL symphonies had minuets. I don’t know if they did or didn’t; but in music we tend to focus on the greats and not the common practice.
Beethoven evidently found the Minuet a bit too flighty for his works and switched over the the Scherzo not only in symphonies but other works as well.
By the time of the symphony, the direct connection with dancing was long gone (see comments on Bach above). I’m not sure what you’re getting at about symphonies being peformed at social gatherings rather than concerts. You may be thinking more of chambre music or divertimenti (like Eine Kleine Nachtmusik by Mozart). I think by the time of Haydn and onward people were actually listening to the symphony at symphony performanes. You might be conflating what happened at opera (where people did visit during the “boring” recitatives and then pay attention for the arias).
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Here is another - why do classical minuets include Trios? That would seem to argue against the hypothesis that dance was involved, since the Trio would screw up the dance real good
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The addition of the trio to the minuet goes way back to Lully. Originally they were trios, but then their orchestration got expanded but the term stuck.
Would it? Would it be all that different from switching to dancing to another piece?
Two old Southern Belles were rocking on the veranda. One says to the other, ‘Do you remember the minuet?’ The other replies, ‘Hell, I don’t even remember the men I f*cked!’
But the third movement, after the trio, switched back to Minuet form. Kind of like when I was in college in the early '70s. You’d be dancing to a song, then they’d break into a guitar solo or something and you’d go “what? how do I dance to this?” and then they’d go back.
I’ve listened to the Greenberg Great Orchestral Works and Beethoven Symphony courses and I don’t remember him giving the origins of the minuet movement, though he did cover the structure of them. He certainly did not say anyone danced to it, and knowing him he would have.
I’d guess that it was a case that composers liked to create new works within the bounds of quite rigid structures - which Beethoven exploded.
The trio is generally in the same time and tempo as the minuet, so if you wanted to dance to it, there shouldn’t be a problem. It’s usually just in a different key and (originally) a contrasting orchestration (being, generally, composed for a trio).
Fun Fact: The Russian word for “blowjob” is minyet (stress on the second syllable). I nit you shot! 
That was to get you off the dance floor, over to the bar, to buy a drink, to boost the bar profits, to pay for the music. If your fans didn’t buy, you didn’t get booked to play.
I don’t know if there were similar commercial imperatives for Symphony composers.
Would it? Jotas, a form (or rather family) that’s considered traditional in a lot of Spain, often involves changes of tempo and/or grouping (pair to double pair, back to pair, then the whole ring, etc.); for a while, mixed forms such as Astráin’s waltz-jota were quite in fashion. So long as you know the change is there, it can go perfectly smoothly (not the best image, but the carriers are better dancers than most of the ones I can find in youtube; the same song is usually danced at the end of every gigantes y cabezudos meetup, with dozens of dancers).
The practice predates the Viennese school by almost a whole generation. From what I’ve read, symphonies with minuets originated in Mannheim, when Haydn was still a teenager.
I’m interested in the early stages of the symphony; not the late 18th century, but the 1750s. The period, in which three-movement symphonies were the norm, and four-movement symphonies with minutes were seen as innovative.
During much of the 18th century, symphonies were treated as disposable decoration for social events. They were commissioned for specific events, performed without prior rehearsals, and then never performed again. This practice only started to change in the 1770s.
The Trio is essentially a second Minuet. As fachverwirrt mentioned, it’s in the same metre and tempo as the Minuet. Originally, composers just wrote a second Minuet, embedded in the first one. You can often see it in Bach’s Suites (Menuet I / Menuet II / back to Menuet I).
I guess that the secondary Minuet evolved into the Trio simply by giving the orchestra a rest, and just letting the first violin play together with another solo instrument and the continuo. It provides a nice contrasting effect, and it’s more elegant than just writing one single Minuet and playing it three times in a row, each time with different instruments (which was also often done - allegedly).