The closest thing to a genre name for this is "Sturm und Drang" ("Storm and Stress"). This is dark, violent music. It started around 1800 and followed a literary movement of the same name. (Any novel that starts out "It was a dark and stormy night" is probably Sturm und Drang.)
Haydn experimented with it once or twice. Listen to the first movement of the "Farewell" symphony. Beethoven was all about Sturm und Drang. For the fast violent stuff there's the Appassionata sonata, the last movement of the Moonlight, and the first movement of the 5th symphony. For the slow depressing stuff there is the slow movement of Symphony number 7, and the slow movement of the sonata number 7 Op. 10 number 3 which is absolutely the best music to play when there's a violent thunderstorm outside and they have locked you in the room at the top of the tower because you KILLED YOUR LOVER and ATE HER HEART RAW!
This kind of music continued well into the twentieth century. For the violent kind: Mendelssohn's First Symphony, Wagner's Ride of the Valkyries, and the piano concertos by Dohnanyi and Rachmaninoff.
For the slow depressing kind: The third act of a Wagner opera frequently opens with things-are-the-worst-they-could-ever-possibly-get-and-getting-worse music, notably Tristan und Isolde and Meistersinger. Liszt's "Funerailles" for piano solo is another one, alternately depressing and violent. See if you can find Petrassi's "Coro di Morti", a very depressing piece for male chorus, and Gorecki's Symphony number 3, "Symphony of Sorrowful Songs". You'll need a sound system with good strong bass for that one; it opens with a slow eight-voice fugue for double basses.
Mahler's "Tragic" symphony (number 6) depicts a man who is smashed to the ground by a hammer-blow, not once but three times. The last movement of Tchaikovsky's "Pathetique" Symphony (number 6) is as dark as music ever gets. Oddly prophetic too -- you can look at it as a depiction of his entire life. The premiere performance was a great triumph and corresponds to the end of the third movement. Shortly thereafter, something totally unexpected happened and his life was ruined, just as the fourth movement predicted! How did he know?
Shostakovich's Fourth Symphony -- listen to the whole thing and when you get to the last three minutes, turn the volume up loud.