I can't really answer the first part, but I can talk about the second: no, pop and rock don't come from classical music. There was no transition from classical to popular music.

Classical music is a musical tradition, it's not just one thing. It's music that has had some things in common (a writing system, efforts to teach it and preserve it, financial support from the Church/the government/the rich, it has been associated with the cultural and intellectual elite), many completely different kinds of music are put in that category because they come from this lineage. I am not trying to say popular and classical music have been completely separate, there has been some exchange between both, but it's not like there was only classical music in the past and then popular music was invented, or that it was built from parts of classical music.

The scope of this question is very large! Each of the modern music genres you mention has a rich history and a rich set of influences, and explaining them all would take some time. I am going to do so quite broadly, without much subtlety, but if I were to pick the three most important differences between, say, Beethoven's fifth symphony, and the Beatles doing Chuck Berry's 'Roll Over Beethoven', they would be:

Musical structure: European classical music like Beethoven is structured very differently to a pop song. A symphony or a sonata is supposed to have particular musical structures that the audience expect - movements, recapitulation, motifs, etc. - and these structures are not present in pop songs, which audiences have different expectations for (verses, choruses, bridges, refrains, and more recently, 'drops'). The obvious pre-20th century source for pop song structure is European folk music; there are plenty of examples of European folk music which alternate between non-repeated sections and repeated sections, a little like a verse and a chorus in a standard pop song. And broadly, the singing styles in modern pop are more closely related to the singing styles in folk than they are related to the operatic singing styles most closely associated with European classical music.

Rhythm: A Beethoven symphony tends not to be anywhere near as strongly dominated by rhythm as a modern pop song. The obvious pre-20th century sources for the way that modern pop uses rhythm are the West African musical styles that were brought over to the Americas thanks to the slave trade. These came into popular consciousness in the West via strongly African-American forms of music such as jazz, and the blues.

Technological advances: European classical music only exists in the form it does because of particular technological advances - a piano or a violin is a complicated piece of machinery. And technology has advanced further since Beethoven's day - he likely couldn't have imagined the ability to record and listen back to a sonic event, let alone electrical microphones, synthesisers, and computer programs like Garage Band or ProTools. And the possibilities in a piece of music change dramatically when it is transmitted through hills and valleys on a groove on a 78rpm record which are the same every time, rather than through a written musical score which is played by a particular, usually standardised, set of musicians. Part of what makes pop music today sound different today to European classical music is that there is a much stronger emphasis on unique and interesting timbres, whereas European classical music used standardised sets of timbres - say the typical symphony orchestra, which has delineated sections of musicians playing particular kinds of instruments which usually have specific roles. Part of the appeal of, say, Michael Jackson is the tone of his voice, and the way that he sings the words and notes he sings. And these are things that are hard to convey using a musical score to someone who has never heard Michael Jackson before. And so because European classical music in the 19th century was fundamentally based around a written-down musical score, it can't convey the sound of Michael Jackson's voice or the feel of a rhythm, but can convey rich harmonic ideas and melodic ideas reasonably well. All of the genres you mention - "rock, pop, hip-hop, techno" - are genres which exploit the artistic potential of timbre in ways that would be unimaginable to Beethoven.