Four "tamable" robot variants can be found in Tangerine District on planet RockTropia. 

Ladybug Crickey can be found in "Torvald's Bog".

It belongs to the "Defensive" class and its rarity level is "Legendary".

The Black Beagle is a member of "The Pack".

It belongs to the "Aggressive" class and its rarity level is "Rare".

I had already encountered the Phoenix Rooster once, but my pet had lost the battle. I then decided to spend a lot of time with fights at low level (Farmyard Frenzy) and finally the Phoenix Rooster appeared again. This time I was prepared to win!

The Phoenix Rooster belongs to the "Balanced" class and its rarity level is "Legendary".

I just discovered a Venomander!

Venomanders are born in poisonous, foul smelling marshes. This stains their skin and makes them smell awful. They belong to the "Attuned" class and their rarity level is "Epic".

I have just acquired the Monria stable which was put for sale on the auction. This stable is now up and running.

The Pet Center Manager currently offers following buffs:

I hope we will see new developments of the taming system in the near future, with the addition of new features to the stables.

Pygmy Mountain Gorilla is a level 15 creature that is found on Secret Island. It is not especially difficult to tame. The point is that it spawns in a wave area, which makes things a bit more difficult. It is very important to stay just outside the wave area during the hunt in order to avoid Mountain Gorilla waves to start. The best position is [38184, 19467] and you need to be equipped with a long range weapon to be able to tag any spawning Gorilla. Then you have to kill regular Gorillas over and over again until a Pygmy appears. The regular Mountain Gorilla spawn is extremely small, with two Gorillas occuring at the same time at most. All you need then is patience...

The problem with the tamable papoo spawn south of Crystal Center seems to have been corrected. Yesterday I went to Next Island and found Adaptable Brown Papoos that can be tamed. These are level 1 creatures and are therefore very easy to tame. Once tamed, their name is changed and they are called Lofven Papoos. They cannot be exported to any other planet so you can only summon them on Next Island and Secret Island for the time being.

Edit: The issue is not solved. Most spawning papoos are immediately killed by the nearby turret. It is only in exceptional situations that some of them spawn outside of the turret range and can survive.

I have found and tamed a Pygmy Kong. Pygmy Kong occurs occasionally among regular Kongs on Secret Island. This is a level 36 creature, and the most difficult I had to tame so far, much more difficult than Hogglo Pygmy. I am now level 40 in the Animal Tamer profession and level 50 in the Evader procession, and the process of taming a Pygmy Kong took me more than two hours. Run RiverRun Run was present during the first hour, and could provide assistance with healing and moral support. 

The whip I used - a Dominax Original Boiga (L) - lost more than half of its TT value during the fight. I also used a Portable Repair Unit (L) to repair my Angel armor regularly for greater comfort. Pygmy Kong has a very high spirit, and it takes time to lower it sufficiently and make an attempt to tame this ape. In my case, two attempts were necessary. Maybe the Artemic Ring (L) I wore helped a bit.