I'll break down the trash pulls for you in a semi-speedrun (can be done in AF with HQ Amber/Militia and onward):

First pull is grabbing the 2 tonberries then running to the 2nd room and grabbing the birds. Kill all that, run up the stairs, spawn the bugs, and AOE 'em down. Grab the next trash pack (birds + ton I think), kill all that, then move into the room with the trap chest and grab everything inside and move them to the bottom right corner with DPS and heals on the middle right side. This allows them to avoid the AOEs and not get caught by the Stalker (he'll always go left first).

After that, pull the next pack, AOE, next pack (tons first then move the rest into the boss room) and you're on to the first boss.

Next pull is Ton -> Pugils -> Ton. Then go up the stairs, kill all the stuff, move into Stalker Hell. This room can give you a bit of problems depending on where the Stalkers are when you're pulling, but the safest method is to kill the Bones, Flash the spirits, Provoke one of the Tonberries guarding the boss door and try to kill them all on the stairs. After this, there are 2 mobs (skeleton and ghost) near the handle you need to turn along with a Stalker patrolling nearby, so what I'll typically do is grab them with Shield Lob/Tomahawk/Provoke, pull them near the boss room, have a DPS turn the switch (Flashing/Overpowering the mobs in the meantime so that the healer can heal), then run into the boss room. At this point you can either lock them out by standing right in front of the purple line and having someone else pull the boss for you, then grabbing aggro with Tomahawk/Lob and waiting for the 15s when the mobs will get locked out... OR you can just kill the mobs inside the boss room before pulling the boss.

Dodge Fire by running through, AOE down mobs, don't die, onto the next trash. 3 spirits are next, one of which will cast an AOE (yellow one iirc) - instinctively, after these are dead, you'd think to just run up the stairs, kill those tons, and grab the chest. However, running up the stairs triggers a Stalker to spawn - so what you'll want to do is (either while the spirits are still up or after they die) Provoke one of the tons down the stairs, kill them, THEN go up the stairs to grab the chest. After that it's 2 simple pulls, you can either pull one to the other or kill one at a time. Tonberries always take priority if theres any mobs left over by the time the Stalker catches up simply go into the boss room and kill them there.

Tonberry King, don't die, gg.