'sup Anekei =D
I'm an assault whore, so I figured I'd give you as much info as I can remember about these missions.
Supplies Recovery
From what I've heard from other people, this can be a real pain if you don't know the rule about the blue mages. As roadkill mentioned, if the blue mages kill the mob you're working on, even if you've dealt the majority of the damage, they'll get the credit for the kill.
As a solution to this, people have said that they let the blue mages do most of the work, then unload on their target with weaponskill, hard nukes, and whatever else you've got in your aresenal. Either that or kill your targets fast, but I doubt killing imps fast is as easy as it is said. I never did this one, so that's really all I can tell you about it, sorry ><;
Apakallu Seizure
I can't tell you how many times I almost got an ulcer from the stress this caused from incompetent party members who just don't listen. Since you've linked to the wiki pages, I'll assume that you've read the page, though there is some information that IMO, it lacks (though I've not read it in whole).
Here's where the ulcer comes in - have your other party members stay the **** away from you while you're guiding the apakallu. While guiding it be sure of four things:
A) Party members are not in your path
B) Monsters are not in your path
C) You do not run too far ahead of the apakallu, go slowly to be sure it can keep pace
D) Be aware of any obstructions that may block the apakallu from following you, such as a steep incline that you may be able to climb up.
Following this, you should be able to get at least one apakallu to the goal, which is all you need. Avoiding conflict with the orobon NMs is a good idea for time conservation.
One last thing to note is that if an apakallu runs away, don't chase after it because that's simply time that will be lost as it will run from you too (unless SE gimped this assault mission with the slew of other things). Find a different one, or wait for the original apakallu to return to its normal area where it just wanders around (this also takes time but is good to note). Send me a tell in game if you want any help on that or any more info that I failed to provide
Edit:
In addition, it helps if you make macros for your emotes and trades, rather than taking up time typing them. You may also wanna make one that tells your party your current position and to gtfo. Kiz is right about the hamzi, I just assumed you knew of it since you linked to the wiki page.