Each pet description is vague, and may not be 100% accurate.
Pets will pick-up elements only twice per game.
Pets found in the Shop
Green Slime

Purple Slime

Blue Slime

Goblin Shaman

Skull Mage

Magma Golem

Stone Golem

Santa Claus

Snow Man
-Point and Code 1.2 times gain [Update as of 2/13/19: Point and Code rate boost decreased by -50%]
-Guild Points 1.2 times more
-Will pick-up elements coming by

Pets not found in the Shop (available via GEN coin redemption only):
Beagle
- Increase Move-speed for 4 seconds
- Will pick-up elements coming by

Red Dragon
- Fire Sphere attack, hits one monster, 20 damage
- Will pick-up elements coming by
- Increase move-speed twice per game

Leo #1
- Critical Damage for 1.5 seconds
- High Level Element pickup ability, will pick-up elements more efficiently

Leo #2
- Special skills cannot be activated on you for 6.5 seconds*
- Will pick-up elements coming by

Magic Penguin
- Increase Magic Card activation chance by 10%
- Magic that fails will refund MP*
- Will pick-up elements coming by

Mini Pig
- Will pick-up elements coming by
- Elements picked up will double their amount

Pet Stealing
Some pets can be stolen in PvP game modes (such as duel, teamplay, crystal capture, etc.) by hitting them with melee attack and depleting their HP to 0. However, ownership of the pet can be reclaimed by original owner or stolen by another user who reduces the pet’s HP to 0.
With the exception of Snow Man, all of the pets purchasable in the cash store can be stolen. Snow Man and the special pets obtained via GEN coin redemption cannot be stolen.
Pets that can be stolen: Green Slime, Purple Slime, Blue Slime, Goblin Shaman, Skull Mage, Magma Golem, Stone Golem, Santa Claus.
Pets that cant be stolen: Beagle, Red Dragon, Leo #1, Leo #2, Magic Penguin, Mini Pig.