1) Yes, a Diregraf Captain
's ability will trigger if both himself and other Zombies die simultaneously. Creatures die at the same time, so abilities that rely on it still go off. Also, the Captain says, "Whenever another
Zombie dies," so no, he cannot set off his own ability.
2) Yes, both abilities would go off if a Gravecrawler died. Each one is activated by the "Whenever" trigger, so they both go on the stack and (assumedly) resolve.
3) As long as the Zombie goes from the Battlefield to a Graveyard, then yes, even Sacrificing a Zombie will trigger the ability.
4) No, you cannot sacrifice permanents, shuffle your deck, or discard cards without an effect to make it happen ( Altar's Reap
, for instance, would allow you to sacrifice that Gravecrawler, or more commonly, Mortarpod
5) Yes, you may pay
to replay the Gravecrawler immediately after sacrificing him to Grimgrin; however, note that I said "replay" there, as it is important. Gravecrawler's ability allows him to be replayed
from your Graveyard, meaning you play him as if just cast from your hand. This means that it can, indeed, only be done at Sorcery speed, and that yes, he does have Summoning Sickness.
Hope this helps!