I'm a little bit lost when pet hatching and I can't seem to figure it out.
Is there any way that I can make sure that when I hatch, I am able to keep the original look and the card of my pet, or is it constantly random? Because I made a hatch earlier with the hatching kiosk and I ended up with the look of the pet I hatched with and I lost my card, when I wanted to keep the card and look of the pet.
No, other than some exclusive pets that won't transfer from player to player, both the pet species and talents are chance. It usually takes multiple attempts to get the right pet with a given talent, and many sucessive hatches to get a specific pet with an exact set of talents.