Pets are not items and will not be in your bank, pets will appear on the page the pet can be acquired in in the mini-HUD.
The total amount of Pets and all pets acquired can be seen in the Pets section of the Completion Log.
Once a pet is acquired it is permanently active and cannot be turned off.
It is possible to acquire a Pet Offline.
The Chance to acquire a Pet on any given action of a Skill is:
[math]\displaystyle{ \dfrac{\text{Time required for action(s)}\cdot\text{Virtual level of Skill}}{25\;000\;000} }[/math]
For combat an action is an attack. For Slayer, an action is a kill that counts towards the player's current Slayer task, with the player's attack speed used as the action time.
All pets except Larry, the Lonely Lizard and Golden Golbin are acquired this way.