I've always thought of Doom as having sort of an association with "law"--not the kinds of laws that Terezi has a fixation upon, but the laws of reality--this is an obsolete etymological usage of 'doom'. So from this perspective, "death"'s relation to doom is not "being doomed to die" so much as "the laws that dictate the closing of loops and the ending of processes". As a computer science guy, Sollux would surely be intimately familiar with all of this.