Concerning the end of the world, I remember "The Flames of the End" on the last At The Gates album, it made me feel like I was listening to the world's collapse, probably the most intense instrumental I heard since years.
Concerning the funerals, I would have chose.. "Hallelujah" by Leonard Cohen (first choice for years), "Everything" by Norther, "PMIB" by CoB, "Under the Waves" by Swallow the Sun, "Still Loving You" by Scorpions, "Nothing Else Matters" by Metallica when I was 14, "Everybody Hurts" by REM, "How can you Mend a Broken Heart" by Al Green, "Sacrificed Sons" by DT, "Where At Night The Wood.." by Empyrium or even "The Forthcoming Answer" by Your Shapeless Beauty (my favorite metal song) but my decision nowadays definitely goes to Opeth's impressive "Epilogue".
I think about this question many times a week, then the songs change a lot. I like always be searching for the most depressive and sad songs ever.