1) "1/3/6 VPs for each mountain area in the clan territory with 1/2/3 brochs" -> Do the mountain areas need to be completed? The word completed is not included here as it is on other scoring tiles.
No, they do not need to be complete. Neither the art nor the text indicate "completed" areas.
2) "5 VPs for the player with the most tiles with barrels of whiskey in their clan territory ..." -> Again, do the areas need to be completed or the tiles just need to exist in my territory? Is "my territory" simply the set of all of my connected tiles?
Ditto. Your territory is all the tiles connected to your castle tile (whether connected by road or not).
3) "3 VPs for each water area in the clan territory with at least 1 ship and at least 1 adjacent lighthouse" -> Once again, does this territory need to be completed? It is not stated explicitly as it was on other scoring tiles.
Don't mix "area" with "territory." Your areas do not need to be complete for this scoring tile.