For the expedition part, do I need to clear the minion or I get the same progress if I just rush the boss?

It based of your ratting and the difficult so if you kill zero trash mobs and get an F you get very little to almost zero progress to the item. But in order to get an S you only really need to kill 40 or so mobs (idk the exact # but it around 40) as you make your way to the boss and not die once.

Also, make sure your Rare Drop Boosters are active when the Quest ends; I once had mine run out during the boss fight, and I only got a couple percent instead of the 15% I would have gotten if they hadn't expired.