If the card effect is in the description before the list of cards it combos with, then you don't need the combo cards for that specific effect. Any effects listed after the list of combo cards will require all cards listed in that combo to be equipped at the same time in order to receive that bonus effect.
As far as the gift boxes vs obb's:
The Myst Case monster drops obb's occasionally.
The Myst Case card gives you a chance of getting a gift box drop with each monster you kill.
The effect of the Myst Case card is not combo-dependent. In fact, the only card in the merchant combo list that has some of its effects which are dependent upon having the full combo is the Hylozoist card. All other cards in the combo (Raggler, Myst Case, Baby Leopard, and Zipper Bear) do everything in their description without the combo.
It is also worth noting that equipping two Hylozoist cards will not double the combo effects (+20 luck instead of +10, etc, etc). The regular effects (% chance of polymorphing monsters when you hit them) will double, but not the combo effects since you can only wear 1 of each of the other cards (except Zipper Bear since it's weapon-slotted).