I think the popularity of the type of music that you're referring to had less to do with the age of the people listening and more to do with the social circles people ran in. (Keep with me here for a moment.) There were vernacular musics (folk music, etc.) back then, just as there are now (popular music and jazz, etc.). But unlike now, where people go to a concert at the symphony, people didn't do this in huge numbers, at least by our standards. Heck, the idea of the public concert didn't even really hit until late in Bach's time. Previous to that, concerts and musicians were confined largely to sacred and courtly settings. (Composers in the Medieval period of music began to work in vernacular idioms, especially the poetry, into their music...people like the poet/composer Machaut, in particular--see Taruskin, Oxford History of Western Music, v. 1.) These private courtly settings included concerts by court orchestras and musicians and even operas. The Esterhazy family is a famous proprietor of music, including supporting Haydn for a time (there are lots of interesting stories about his tenure there). As the public concert popularized with people like the Mannheim Orchestra in Germany and in Paris in the mid-18th century, it became a social function. And something worth noting is that this was the popular music of its day. "Classical" music as a category is a fairly modern conceit. Like I said, there were vernacular musics, but if you're thinking about music that would have had large circulation, then that would have been what you're calling classical music. It wasn't until the late 18th century and 19th century with the rise of a middle class and salon culture that any music making was done on a mass market scale. Lots of chamber music from around Mozart's time was written to be playable by amateurs. Frederick the Great was an accomplished amateur flutist and composer and set some precedent for this trend, though that trend is an incredibly complex thing that involves a lot of different factors. For evidence of many youths participating in music, you can look at Vivaldi, who for a time led one of the best youth orchestras in Italy, possibly the world at the time. Granted, those girls were orphans and likely didn't have much of a choice in their day to day activities, but they did it. As women became more educated, the girls were expected to play music at a competent level in the salon, singing and/or playing piano, etc. Fanny Mendelssohn (Felix's sister) and Clara Schumann (Robert's wife) are examples of girls who did this and exceeded the expectation to become world class musicians. And there are plenty of examples of child prodigies who went on to long careers in music, so presumably they didn't hate it, though people like Liszt had mixed feelings at different times (see Allan Keiler's psychological work on Liszt). Being a boy singer in a British choir has long been an enviable position.

That's sort of a sideways answer to your question, but it gets at the idea of classical versus vernacular music before the twentieth century. There are lots of examples of music that would have had some currency long ago, things that have survived in some form or another. Music wasn't notated outside of sacred and aristocratic settings until relatively late, but things like "Hot Cross Buns" and some form of "Ring Around the Rosie" come from the 14th century, possibly (hard to know exactly). So, yes there were other things that were popular, at least enough to have survived. Lots of other folk music from any culture has survived, too. Dance music was very popular, as were other functional musics, and I would guess they would have been popular with young people. Composers even set these in more refined styles, like Brahms's famous Hungarian dances and Liszt's rhapsodies (exotic music was a popular tool and is rooted in non-classical origins) and anyone who wrote things like waltzes for concert orchestra. Christmas carols (since its timely now) are good examples of functional music with great staying power. Oral transmissions are incredibly important in all sorts of disciplines!

As a final thought, many classical musicians were essentially rock stars of their day. The 18th century saw the rise of the virtuoso. People like Mozart, Hummel, later Paganini, Liszt, Chopin, Alkan, and many others in the 19th century. So far as I know, no vernacular musicians gained their level of fame.

Mods, this issue is so broad that it's hard to provide individual sources for this information, since it's contained in more or less every survey music history book, but OP if you want to read up on this, look for Grout's history of western music, which touches on these issues sporadically.

I won't get into whether or not "most" of today's youth dislike classical music. :)

There is a small problem with your question, which is that the idea of "young people," teenagers, tweenagers, all those troublesome years of human development that we find so distinct and natural now as categories of identity, did not exist. Teenagers did not have a distinct culture or cultural identity in societies where the composers you are mentioning existed. So it's sort of fruitless to consider what they might have thought about art music of that period, in the same way it's fruitless to ask "What did Seattle tech workers think of the Gold Rush?" Yes, there were of course 17 and 18 year old human beings in the time period, but they were not distinct from adults in their society. There's a couple of threads around here on the "invention" of the teenager, but safe to say it didn't happen until a while after your period. So you can't make a direct line comparison to kids these days and their tweeting Arianna Grande videos to whether or not they liked Mozart.

That out of the way, I can give you a bit of an answer on the ages of people who consumed music though. Most of the social spaces where art music of that period was enjoyed (like the opera house, or house parties) were not spaces where children were taken. So like the idea now where you stuff your children into velvet dresses and little suits and then force them to sit through The Nutcracker, that didn't happen. But younger adults (depending on social class) did go to these spaces, as they were adults. They probably enjoyed it about as much as any other random assortment of people.

You might be interested in the ages of people working in music? During this period people started working in music at a pretty young age, professional opera singers started singing much younger than they do now. My username, Caffarelli, started singing in opera when he was 15. Farinelli debuted as the same age. Much later in the 19th century, the famous Adelina Patti made her debut at age 16. The famous 18th century flutist J. J. Quantz started working as a court musician around age 16 or 17. They would have been considered adults at that period though. All of them studied music as children. So some young people definitely liked music enough to pick it as their job for the rest of their life.