All the pets you can obtain from drops/doing something.
The rest are obtained from lootboxes/quest-coins/raffles.
Meth Pet (The Breaking)
How to obtain: Roll the number 101 out of 1-20k on Meth Cook.
Special: Increases meth yield from the "Meth Cook" by 5%
Hobo Pet (Hobo)
How to obtain: Roll the number 1 out of 1-20k on trash from dumpsters.
Special: Throws bug-bait at passerbys. Cooldown of 30 seconds.
Mining Pet (Miner)
How to obtain: Obtained randomly from mining (not sure of chance).
Special: You dig rock nodes 30% faster.
Fishing (Fish)
How to obtain: Obtained randomly from fishing (not sure of chance).
Special: Time to catch a fish reduced by 25%.
Hitman Pet (Hitman)
How to obtain: Roll the number 47 out of 1-20k on hit completion.
Special: You gain +2XP for every hit you complete. ICU Talent updates 25% faster.
Dragon Pet (Dragon)
How to obtain: Roll the number 1 out of 1-800 when destroying a dragon egg.
Special: Shoot 3 fireballs when entering combat. 180 second cooldown.