How do you complete these? For example, I’ve completed [Honey-Sipping Heiress] level 2-7 at least three times, but it stays open on my Ventures Tab.
Good question! Many storylines stay displayed in your Story or Venture even after they’re completed because of the way Fallen London codes things, and because events in the game can continue to be affected by a story after you finish it. Example: there are two different endings to "A Finder of Heiresses." One ending results in "A Finder of Heiresses 2" being listed in your story tab, the other, "A Finder of Heiresses 5." You can’t go from "2" to "5," but later cards you draw will be slightly different depending on how you ended the story. So don’t worry about trying to "get rid of" a storyline. Your Stories tab does become quite crowded, but that’s part of the joy/insanity of the game…
Routes aren’t stories that you complete. That section simply shows you which of the harder-to-find places in London you have discovered the way to. You will unlock more as you play.