Sonata and Waltz are forms, the closest modern equivalent would be calling something a ballad or a rocker. That doesn't quite translate though, because the Sonata and Waltz had a very specific structure. Still, if you told someone you heard a new Waltz you really liked they would immediately know what you were talking about, a piece of light airy music with a slightly up beat tempo in 3/4 time meant to dance to.

A lot of these pieces did have names in addition to the title announcing their format. The names aren't used as much because musicologists (music historians) have a system for cataloging music (one started by European composers in the 18th century) where they assign an Opus number to each piece, which places them in a sort of chronological order by performance or publication date and describes each according to type. Sonata, Symphony, Waltz, Concerto, Opera, Mass, Etude, and Cantata are some of the more common forms. Some very famous composers with a large library of compositions have unique formatting as well, anything by Bach is given a BWV number instead of referring to it as Opus #, Mozart's music is usually given a KV number named after the first person to catalog his large body of work. Mozart was a major pop star in his day, unlike Bach who became much more popular after he died.

Some of these are almost always known by the name, not the Opus Number. Operas and Cantatas are referred to by title, not catalog number. Others are known only by the number, or by both name and number. Beethoven's 9th Symphony is referred to by number, or as the Choral Symphony, or by it's most famous section which is based on the poem Ode to Joy. His 6th Symphony is almost invariably referred to as the Pastoral Symphony, his 5th Symphony, written at the same time as the 6th is universally called Beethoven's 5th. Mahler's 2nd Symphony is almost always called The Resurrection Symphony, Dvorak's 9th the New World Symphony (it is actually titled by the composer as "Symphony No. 9 in E Minor, From the New World", he wrote it in New York while studying American Indian and folk music, teaching American students and trying to build an American School of composition).

So to wrap up that part of the question, everything classical gets an Opus number and is cataloged by genre, but many of them have other names, some are known mostly by the name, others mostly by the catalog title.

As far as famous singers and songs definitely yes. The most common type is the tavern singer, the equivalent of the modern bar band. J.S. Bach frequently played in taverns himself, actually. But in larger cities Opera singers, concert singers, and solo composers were quite famous, sometimes became very wealthy and very popular just like modern performers do. Opera singers, Castrati, violinists and Piano soloists were particularly famous in some instances. A moderator specializes in Castrati, you can find some great info on them by searching the term in this subreddit. Niccolò Paganini was an exceptionally famous solist, Franz Liszt was treated a lot like the Beatles were in their heyday according to contemporary accounts. Opera in particular created big stars. In it's heyday Opera Houses in places like PAris and Vienna were followed the way American Idol was for the first few seasons, or like the big summer movie blockbuster season is, with every announcement pored over, every concert sold out, and every performance analyzed and debated incessantly.

As far as popular songs go, there were different ways to evaluate them because there were no billboard charts or anything like that before the 20th century. Still, popular songs were common (people loved music if anything even more than they do today, at least in Europe and pre 20th century America where owning instruments and playing for entertainment was much more common than it is now). Popular music tended to have a very long shelf life, and people sang standards more than originals for the most part during live performances. Popular songs were often borrowed by composers for use in more formal works. John Taverners Western Wind Mass is a great example of this, borrowing it's words and melody from a popular English song called Westron Wynde. The big solo songs in Opera are called arias, and favored ones would go into the singers concert repertoire. Greensleaves, still popular today (and the genesis of the guitarists Greensleaves game) was a simple folk melody.

I could probably list songs for an hour here and not cover much of this very broad territory, but you have heard more old popular music than you probably think, whether it's older church hymns, folk songs, Christmas songs, or bits of opera thrown into movies here and there. If you would like an intro to just one segment of this there is a very popular cd by Luciano Pavarotti called Songbook, on it he sings his favorite Italian standards, they are the kind of old pop music, and Pavarotti himself is a great modern example of the kind of very famous singer you are asking about.

Also, why didn't they name their songs original names? Why are they all sonatas and waltzes and stuff?

Since you already got an answer to songs, yes, famous singers abound, in fact, almost always a singer would be more famous than the composer. People came to see Handel's primi uomini more than they came to see Handel! If you give me a time frame and nationality I can probably rattle off the famous singers for then and there but as it stands this is too broad for me to say anything intelligent about!