Good heavens. There is a tremendous variety of "morbid" music -- or music regarding death in one way or another. Several of the above posts have mentioned excellent examples, but you mentioned "other death-related music" -- other, I presume, than piano music. You did not mention choral music, however; a great omission!
I can think of one odd example which has always affected me, somehow -- the death of Jean Valjean from Les Mis. A musical, incidentally, which has many of the characteristics of grand opera, and is worth studying from the standpoint of serious music theory and composition.
However, that aside, as a place to start you should look at the huge array of Requiems which various composers have written. Just for starters, and covering a range of styles: Mozart, Brahms, Berlioz (to my mind quite possibly the single most completely satisfying piece of music ever written by anyone), Faure (totally different, but almost as good), Britten, Rutter, Dvorak, Verdi.
The list of motets which might be put in is staggering. Ranging from Palestrina to Willan and Rutter and Thompson. I'm not even going to try to put them down... but any really competent Minister of Music should be able to haul a dozen or two out of his or her archives for you to look at.
For some reason (I don't know why) I have always associated Bach's towering Prelude and fugue in E flat major, BWV 552 ("St. Anne") as usable funeral music -- and have used it that way, but only for major funerals. The fugue is a triple fugue, and is simply incredible.