As usual, the answer is complicated, but it's more simple than other complicated answers. The complication has to do with time period; when we say Classical music, it ranges hundreds of years that are categorized by styles that you mentioned, Baroque, Classical, and Romantic. While they are categorized by year (Baroque lasting from roughly 1600-1750, Classical from 1750-1800, and Romantic from 1800-1900, all being rough), they are different from each other.

The early Baroque is mainly funded by rich merchants but slowly spread across Europe till the style is widely diffused to all of Europe, famously ending with J.S. Bach. By the end of the Baroque era, music was being paid for (the royalty or rich being patrons of the musicians) by Royalty. A composer would either be paid to write a piece or dedicate it to a member of royalty. This would be the main method of music production (within the Classical genre as folk music still existed) until the early Romantic Era (first few decades of the 19th century).

The change came due to a rising middle class. In Vienna, there were a multitude of middle class workers that would play music for themselves. The composer Franz Schubert best exemplifies it, often he would compose music for his friends and play it for them at parties they threw. From here, romantic composers would start to compose music for profit to the middle class; Antonin Dvorak wrote many String Quartets that were specifically written to sell.

Fast forward a few decades from Schubert and you'll find the Strauss family writing waltzes. These waltzes weren't for the nobility but for the public to dance to, which were of course sold so people may play them. If any single composer shows a non-intellectual interest, you might look at Strauss Jr. for composing dance music.

Was classical music the music of the rich and powerful? Yes, mostly.

The composers of the past that we now have in high esteem were working for the church (usually big ones, in cities that were doing well economically speaking) and/or were employed by the rich. This does not mean they were exclusively composing because of money. Well doing composers before the time of Mozart were servants of the courts and had quite a busy schedule composing, performing, rehearsing, teaching and what not to keep the high standards. This musical life was a symbol of status for the patrons, the rich and powerful tried to get the best musicians. They cultivated their musical tastes and frequently were taught by the great masters they hired (this musical life and taste was later used to differentiate old money from new money). Musicians would need their masters permission to get into private ventures. They weren't usually rich (some times their salaries were kind of low) and they were also paid in kind (food, lodging, etc.).

Chamber music was born in the aristocratic homes of Europe...

Then came the independent musician, and they were also frequently dealing with the rich and powerful. Musicians not under full time employment would then teach, play, compose commissioned music, compose music to organize concerts and operas, published music so people could get sheet music and play.

Look at Beethoven's works. A lot of them were dedicated to prince this, countess that, baron von something, etc. He taught at the homes of the rich (status symbol).

/u/caffarelli wrote about who the Opera goers were.

Here's an article on the economics of musical composition in Mozart's Vienna (mentioning some figures about the costs and earning, and the changes mentioned by /u/DonaldFDraper).

The rich patrons were later replaced by the middle classes attending concerts (plus the governments becoming a replacement for patronage). Concert music composers have not been usually too worried about the poor listening to their music.

Was this music for the intellectuals?

A lot of intellectual work was put in their compositions. They were working in a framework with a lot of aesthetic preferences codified as rules (it takes time and effort to learn those). For the layperson, classical music might be weird at times and sound pretty in other cases. But once you know all that is there, it changes how you deal with this music.

It's similar to the excitement scientists, mathematicians and engineers (yes, we do, too) find in the work of the great geniuses. There's elegance in maths just like there is elegance in the way composers did clever things in their music...

Music was not considered an art until about the 18th century, musicians and bakers were not seen as too different tradesmen. Many smart people were into music (some not just in the intellectual side of it, think physics + maths + philosophy + history, but also playing and composing). Rameau (great French composer, the guy who invented harmony) was in touch with several of the great intellectual minds of his time, until he got lost into numerology for "musical" purposes...

Intellectuals were not the target demographic, but they were also not too far away from the rich and powerful. Yeah, it's nice that Kant was dealing with trascendental stuff, or that Goethe was writing great stuff, but that is not going to go too far in a barn in the middle of Povertystricken... They were also trying to get close to where the action happened. Beethoven and Goethe would some times walk in the park. It just sounds weird, doesn't it? Two of the greatest minds ever chatting and having a nice walk.

The music was not usually easy to play. Maybe the Galant style revolution made things accessible again to the non specialist, but by the late 18th century, only the very dedicated amateur players and professionals could tackle the great works.

Composers published music, and this means a lot more people could buy it and play it at home. However, that means they had the money to buy the instruments and the time to play/sing (not to mention the required training). The piano became more and more popular, but it became really accessible until the middle-late 19th century.

Composers also published "lighter" music. Dances, simple songs, etc. Yes, there was a VERY notable difference between the artsy intellectual hardcore stuff and the cheerful casual tunes.

Were common people listening to Beethoven's new quartet in the pub? As far as I know, no, they weren't. People listen to music for a lot of different reasons, and music tends to match the occasion.

What were people used to listen? Well, I don't know much about popular musical (neither contemporary nor very old). I understand these could be close to what was happening back then:

https://www.youtube.com/watch?v=MjNOPDVgjYw

https://www.youtube.com/watch?v=FB4PE0NSkok

Polka was born in the middle of the 19th century. Ländler would be a popular dance from the 18th century.

A LOT of popular music was published in the 19th century. Jingle Bells is from 1855, Stephen Foster is an influential name in 19th century US music.

Check here for some 18th century music.

I mostly know about popular music in the early English Baroque/late Renaissance, but yes, there was plenty. One composer in particular, Christopher Simpson, wrote a lot of pieces that were intended for the average person (maybe not the poorest of peasants, but among the lower middle class at least someone owned a chest of viols, which is the term for a collection of them---treble down to bass, also known as viola da gamba) to get together with a group and play, as well as a method for learning to play and improvise on the viol (The Division Viol, Christopher Simpson). His pieces were very beautiful to our modern ears even, as he made extensive use of major thirds and sixths, which were sort of newly accepted as consonant intervals (instead of the perfect fifth or octave, the preferred interval of the Renaissance). England in particular, as well as the Netherlands, had a lot to do with the switch from modal to tonal harmony because of the focus on major chords.

The Fitzwilliam Virginal book has many settings (versions) of tunes for keyboard, specifically Virginal, which is a type of harpsichord with the string plucked in the middle, giving it a slightly more harplike and less brassy tone. Many of the tunes were popular songs from the time, which are still played by folk musicians and in particular, Morris dance bands---some examples are "Gathering Peascods" a version of which was also set by JP Sweelinck under another name, "Unter den Linden Gruene"--another tune still popular is "the Irish ho-Hoane", or also called "My Thing is My Own", a suggestive and cutely obscene song from the late Renaissance/early Baroque. Nancy Wilson of Heart did a cover of it, it's so well known.

There were also late renaissance musicians like John Dowland whose songs were considered court music, but were known all over Europe, such as Flow My Tears (Pavana Lachrymae) and many others.

Hope this helped. Oh, yes, a good place to look as well is Sweelinck's dance variations, all settings of popular tunes, some quite racy, and as I said before, the Fitzwilliam Virginal Book. Google that and read the history of it, it's quite interesting about Trevian..many of the pieces are from Catholic composers, as Trevian was in prison for being a Catholic in newly Protestant England, which is why Sweelinck was included in the book (also as a Catholic in a Protestant country, the Netherlands). There was a lot of trade and contact between England and the Netherlands, and many of the tunes were the same, or known under other names. (see "Gathering Peascods" vs "Under the Linden Green")

Hope this was helpful. Feel free to ask any other questions. As far as popular music in Beethoven/Mozart's time, you can look to his variations on what we know as "Twinkle Twinkle Little Star" (Mozarts, that is) but what was known to him as "Ah Dirai-je vous, maman", another slightly racy folk song. I don't know much besides that.

Common people were also really into hymns, because a lot of times that might be the only place someone would regularly hear music. Church, that is.

I often hear the claim that the pieces we consider classical music have "stood the test of time" and that what we listen to today from that period is most likely going to be the best of that period and not representative of the average piece, and would contribute to the association between classical music and intellectualism. Is there any merit to this argument?

There were plenty of everyday songs as well, many of them quite bawdy. Some were good working songs, others best for drinking. If you want to hear what some of these (NSFW) songs sound like check out My Thing is My Own, all songs by Thomas D'Urfey who wrote in the early 18th century. Fathom the Bowl is a great rounds song for the end of the evening, likely dating from around the end of the 18th c. or early 19th. Sea shanties like Spanish Ladies were also around (late 18th c.).

It's worth noting that a big part of the partition between modern "classical" and "popular" music as it exists today came about as a result of the advent of recording technology. The complex waveform of a 100+ person orchestra playing the Largo from Dvorak 9 was completely lost by the recording technology of the time while a raucous Dixieland band banging out "When You're Smiling" sounded just fine. The technology required to capture the nuances of western orchestral music didn't come about until the mid 20th century (and even then, the fidelity of said recordings could be considered questionable by modern standards). While this certainly didn't stop folks like Arturo Toscanini (at the helm of the NBC Orchestra) from using recording/broadcasting technology of the time to reach a wider audience, some might say it was too little too late. Jazz music was far better suited for this format, and thus rapidly gained popularity over classical music (Alex Ross writes at length about this in his book, "The Rest is Noise").

Another interesting thing that's worth noting is that the top "classical" or "proper" composers from every generation were often avid fans of folk/popular music and appropriated this material into their music. For example: Orlande de Lassus, a Franco-Flemmish composer from the late Renaissance, was well known for working the melodies of bawdy secular tunes into his sacred masses and motets (there is an entire genre, known as the "parody mass", based around this).

All that said, don't forget that great art tends to follow money. Great composers of generations past (particularly in the Renaissance and Baroque periods) usually relied on wealthy patrons to support them. Every once in a while you got someone like Handel or Mozart who figured out a way to achieve incredible mass appeal and achieve rock-star status while raking in cheddar hand over fist (although Handel got his start simply by having the right wealthy patron), but more often than not, the upper class fueled the creation of great art.