Abilities » Propagation
If I use Exeggcute's "Propagation" Ability to retrieve it from the Discard Pile into my hand, then use something like Ultra Ball or Computer Search to discard Exeggcute, would I be able to use "Propagation" again to retrieve it from the Discard Pile?
Yes, whenever Exeggcute is in the Discard Pile, you may use Propagation during your turn as long as its Ability has not been disabled.
Source: Plasma Freeze FAQ (2013-05-16), TPCi Rules Team (2013-05-16)
Abilities » Propagation
If I use Weavile's "Vilify" attack to discard one or more Exeggcute, can I use Exeggcute's "Propagation" Ability to immediately retrieve it back into my hand?
No, you cannot. Propagation can only be used on your turn before your attack, not during your attack nor afterward.
Source: Plasma Freeze FAQ (2013-05-16), TPCi Rules Team (2013-05-16)