Spoiler Alert! We have hidden this information so that you have the choice to try it on your own, or have the advantage of knowing how the sausage is made...
You do not have to get them down to a certain HP, but you do need to be fighting them to use the Trap. If you use a ranged skill to pull them, and use the Trap immediately while they are running to you, you can capture them before they even reach you.
This quest may let you subdue an extra creature or two beyond the 6 you need, but there is no point as turn-in will take all of your Trapped Jungle Animals and not just the 6 you needed.
Daily Quests are repeatable up to a total (for all Daily Quests) of 10 times per day. The count can be reset to zero by purchasing and using a Daily Quest Ticket from the Item Shop (for 39 ($1.16)† ) or from any Phirius Merchant (for 780 ).
In all other zones they do not reward XP, TP or Gold, only Public Event Points. In all other aspects they are the same as a Daily Quest (except for the infinitely repeatable part and the rewards). Public Event Points contribute towards the completion of Regional Quest objectives, which in turn rewards a number of Energy of Justice commensurate with your amount of participation in the Regional event.
Timesaver!
The following quests have steps, hunting areas, or mob targets in common:
Main Quest Page: Disturbed Jungle Inhabitants
Trapped Jungle Animal ×6
Trap. If you use a ranged skill to pull them, and use the 
Daily Quest Ticket from the Item Shop (for 39
($1.16)† ) or from any Phirius Merchant (for 780
).
Dual-Function Daily Quest Potion.
Old Bag (hence the common sobriquet, Bag Quests). In Redhill Mountains, each repetition of [67][Public] Freedom Fighters rewards an 
Ancient Memento.
