Can't progress the quest "Tango with a Tracker" because of game crash.

[Alhanelem]

1. Started the quest normally.
- Normal
2. Reached the fight area.
- Normal
3. Started the fight normally.
- Normal
4. Game crashed when the boss started a conversation (the boss hasn't aggro'd yet).
- Crashing isn't normal obviously but nearly all cutscene crashes are the result of third party tools. The remainder are just edge cases, like reading bad sectors on your storage device, or other random unpredictable things. Regardless, you have entered the battle area, it doesn't matter what happens or doesn't happen once you enter- disconnects, crashes or leaving without any other players in the battle with you will result in a failiure, which is normal.
5. Restarted the game.
- Okay
6. My character's now outside the fight arena, as if the fight had ended.
- Normal behavior in this situation. The fight did end, because there were no players in the battle.
7. No reward(s) from the quest.
- You didn't complete it, so, this is normal.
8. Quest tracker still on "Tango with a Tracker".
- You didn't complete it, so this is normal.
9. Tried to talk to the quest giver, Despachiaire again. He's now stuck in a normal NPC conversation.
- This is normal because the required time to get a new KI has not elapsed.
10. The key item that as I understood needed to engage the boss fight (Letter from Shikaree X) is now gone from Temporary Key Items.
- This is normal, the KI was lost upon entry.
11. The interactable (Dark Miasma) in Boneyard Gully is now stuck in normal prompt, preventing me from entering the fight.
- This is normal, you don't have the KI anymore, and no other battlefields to enter, so you can't enter.

Hopefully this helps you understand why what you experienced (other than possibly the crash, but it's still unlikely) is normal behavior for the game.

The same thing applies to all other battlefield areas in the game that requires an item to enter. There are exceptions where a KI or item is only used after completion, but the vast majority of fights, the entry item is used upon entry. You don't get a second chance if you crash, disconnect, leave, get warpII'd, get sent to GM jail, or anything else that could possibly result in battle failiure.

Aside from the crash itself, everything you described is 100% normal behavior for the game, in this and in almost any other battlefield area.