The short strings intro is an excerpt from the first movement of Edward Elgar's Enigma Variations. On the other hand, and contrary to a widespread rumor, the piano parts are not "samples of Elgar's Enigma Variations", and you wouldn't find them on Elgar's score: they are Dougan's own composition, played by himself; but this composition is indeed derived from the Enigma variations (especially the visible Theme and variations 1 and 12), and could be considered either as an apocryphal 15th variation, or as Dougan's attempt at solving said enigma, which is the fabled second, hidden theme Elgar said he based his 14 variations on, but never revealed (see details in Enigma Variations's history).
"Clubbed to Death 2"'s classical part is built around Chopin's "Prelude No.4 in E-minor" (from Preludes, opus 28).