Was classical/baroque music the 'pop' music of its time? Were Beethoven and Mozart as loved as Justin Bieber for example?

Interesting that you mention Paganini, because he's really a great example of what the 19th century brought to what we know today as "classical music." But first, it's important to note that our understanding of what is and isn't "classical music" is really based on a modern perspective. As I wrote about in this answer a while ago,, a guy like Mozart who we think of as a quintessential "classical" composer may not have actually seen the same connection between all of his music as we do today. The difference between, say, liturgical music and a serenade is not a distinction made by a lot of listeners today, but would have been immediately recognizable at the time.

So it's worth thinking about where and with whom this music would have been popular, and taking different formats on their own. Some have been hugely popular in their time, others not so much. Three broad categories stand out for popularity among a wider range of people throughout the years.

The first is music written for amateurs. Remember that before the invention of audio recording about 150 years ago, if you wanted to hear music you had to be within earshot of a musician. Many people solved that issue by becoming musicians themselves, and those people needed music to play. Telemann is a great example of a composer who wrote a remarkable amount of music that could be played by amateurs, often with flexible instrumentation, limited range, and suitable for a variety of skill levels. In the 19th century, as the piano became a fixture in many middle and upper-class households, composers like Schubert found a market that was hungry for accessible music.

The second area of relative popularity is the opera. In the 18th and 19th centuries, opera was hugely popular in cities like London, Vienna, Venice, and Paris, and some composers made a killing writing for the stage. Composers like Handel and Mozart who are today known for a broad range of works would have been most recognized for their operas by the general populace in their day. In fact, while Mozart famously died in dire financial straits, his opera The Magic Flute would have helped considerably.

Opera had a reach that went beyond the stage. Thanks to the fact that people throughout history have loved to sing, most popular tunes and arias to come out of the theatres would find their way onto the streets, to mingle with other popular songs. Supposedly, Verdi was convinced that "La donna è mobile" from Rigoletto would be a huge hit, and had the tenor who sang it practice in secret so that the tune wouldn't get out beforehand. The stereotype of the Venetian gondolier singing (among other things) opera arias isn't actually really a stereotype, it was (is) a real phenomenon that ended up inspiring a popular classical form, the barcarolle.

The third area is where we find Paganini. The 19th century brought about the rise of Romanticism, with an emphasis on the artist as a heroic individual. This led to the huge popularity of virtuosi like Paganini, who would tour Europe giving showy concerts to sold-out crowds. These are the closest people to rock stars you'll find in classical music, and the hype and mystique surrounding them sounds very much like that of Alice Cooper or the Rolling Stones. Paganini was rumoured to have sold his soul to the Devil for his otherwordly ability on the violin. He'd arrive to concerts dressed in black, coming out of a coach drawn by black horses that resembled a hearse. As the old saying goes, men wanted to be him and women wanted to be with him. I'm sure he had a few people bowing and saying "not worthy" or whatever the 19th century equivalent was.

There are plenty of other examples of areas where classical compositions became well-known. In an era where church-going and other religious activities were more well-attended than today, liturgical music certainly had a great impact. The Allegri Miserere is one such piece that became quite popular. But the three areas above are probably the three that would most closely resemble what we think of as "popular music" today.