So I actually know a little about this, in the specific case of Beethoven and Mozart. I am not an historian but I have taken some classes in music history. Never answered anything here before so please let me know if I have erred. I can go into more detail on some points if requested.

The short answer is "yes."

It's probably worth noting that in the premise of your question you are comparing 16-17th century London to 18th-19th century Vienna. In Shakespeare's time, opera and ballet were in their infancy, and an orchestra would look very, very different.

For Beethoven in particular, he lived in Vienna from 1792 until his death. A large part of his income was from noble patronage, but he also made money as a teacher, and from giving concerts. In Vienna, "all these theaters give great concerts and performances of the most important ancient and modern music for church and concert hall.... All the great public diversions and amusements are enjoyed by all classes without any abrupt divisions or offending distinctions--in those respects, Vienna is again quite alone among the great cities of Europe." (JF Reichardt, "Briefe geschriben auf einer Reise nach Wien," 1810, as translated by Oliver Strunk in "Source Readings in Music History," p738-30). In Beethoven's time, seats at the theater were not reserved, meaning theatergoers arrived early and had to rush for their seats (the nobility would hire people to occupy seats for them). For the Ninth Symphony at least, posters advertising its premiere were printed and distributed.

The Ninth Symphony premiered in the Kärntnertor Theater, which at the time had a floor divided into forward and rear sections, each with numbered seats (unusual; in most theaters at the time the rear section would have had unnumbered benches). There were five balcony levels; the lower three were boxes that were rented for the season (by the wealthy), and the upper two had benches and standing room, for an estimated capacity of 1000 people. Most accounts of the Ninth Symphony premiere say the hall was quite full. Crowds of the time could be quite raucous, and in Vienna at the time there were laws regulating when applause could happen and prohibiting calls for encores, individual bows by performers, and dogs.

I have less information on Mozart, but like Beethoven, he operated in Vienna (albeit a bit earlier) and in his life was valued more as a performer than as a composer. He also gave public concerts in some of the same theaters.

Source: "First Nights: Five Musical Premieres" by Thomas Kelly, 2000; information on Beethoven and Vienna is in Chapter 3, p108-179. A very easy read with many primary sources if you're interested.

You're asking several different questions, that span a huge length of time (300+ years) and several nations.

As far as Shakespeare, I'm under the impression that commoners could attend, but that's not my area of expertise, so I'll allow the experts to weigh in.

As for Mozart and Beethoven, the answer is a kind of no. While both Mozart and Beethoven held concerts that were open to the public, the cost of these concerts would have been relatively prohibitive for anyone other than nobles, or wealthy merchants to attend. While, theoretically, a peasant could have saved up wages and paid for admission to a concert, it's not likely they would have done so, and it would have been very uncommon.

You may be surprised to know that this question has been asked several times before:

https://www.reddit.com/r/AskHistorians/comments/48pd5h/during_mozarts_lifetime_would_an_austrian_peasant/ https://www.reddit.com/r/AskHistorians/comments/6jccem/im_attending_a_concert_put_on_by_mozart_how_does/dje3n99/

I'll get you to start there, and if you have follow up questions, or want additional information, please comment here, and not on either of the old posts so that I can follow up.