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(ms)}\cdot\text{Virtual level of Skill}}{25\;000\;000} }[/math]