Outside? I fast traveled to Sky Haven and didn't see her, or Esbern.
If you walk up the stairs all the way up once you are inside, you'll see a set of large doors. This leads you to the back yard of the Temple, where the training grounds are. They are normally found there, if not inside the temple.
No, but you can let pathnx live. The journal entry will remain, which is a little annoying, but it's a quest you can complete in the future. Who knows, maybe Beth will make Greybeards evil in a dlc.
Maybe she died during a dragon attack. Oh, the irony!
It's not funny. I used EssentialNPCsBegone and Wars in Skyrim mods, which made travelling rather tricky. On our way to Kynesgrove Delphine died five times. The last one was to a Storm Atronach, try as I might I couldn't destroy it fast enough to save Delphine. Fearing what would happen if I somehow managed to keep her alive to the meeting with a dragon, I gave up and turned her essential again. I lost the fight with game mechanics... It was a sad moment.