The question is: could an animal like this exist in our world?
The creature is viable. I would argue that the tail is the main oddity of its evolution, at some point it or its ancestor must rely mainly on its tail, placing an important evolutionary constraint on it.
The dexterity the creature has with its tail suggests an important brain evolution associated with motor skills. Paired with the fact that it is a social creature, it is not odd that it has complex behavior such as the construction of elaborate nests.
I do believe that the evolution of the tail and the construction of the nest are examples of sexual selection. Elaborate displays of tail movements (such as making knots or twisting it in a spiral) could be attractive for partners; it can be a proxy for display of construction skill necessary for a strong nest. Furthermore, the nest itself could be signaling of social status.
I want to mention the odd fur the creature has, being yellow with black dots. These creatures, being tree-dwellers from the South American rainforest did not compete with other creatures with similar fur. Furthermore yellow does not seem to be a good choice of camouflage for a tree-dwellers creature. I must conclude this fur is vestigial and was not been selected out.
Note: Leopards are not from America. Yes, Jaguars are ground-dwellers from America.
How could its ancestor get to South America in the first place?
Branches of Monotremes separated in the Jurassic, at the time Africa and South America where connected. Therefore, the ancestors of these creatures could have always been in what would become South America.
While in the real world, there are Australosphenida are only found in Australia (hence the name), there is a fossil record of them in parts of South America and Africa.
How could the long tail function and why so?
As per the function of the tail, it would work similar to an arm, but with additional flexing points. Something like a very flexible long finger. It would require a dedicated region of the brain for its control, as part of the motor cortex.
The muscles in the tail would be numerous and small, allowing for fine control of the movement. This also means they are not very strong.
See more below in "The Jump"
How did it learned through generations to build such an odd nest?
There are other social animals capable of passing knowledge by example an imitation. It is interesting to speculate on the memes necessary for the creation of the nests: you need to create the wood structure, which requires picking branches, and you need to learn to pick feathers to stuff it.
In any other context, picking branches and picking feathers are orthogonal. In addition, when building the nest you need the wood structure first, so I would say it evolved first.
The first incarnation of the nest could have been similar to that of birds, with the roof being an evolution of it. The next evolution would be to fill it with leaves for comfort, which is something that may have been an accidental discovery.
By experimentation, they may have started using feathers, which yield better results. However, they did not have the practice of hunting birds (in fact, the linked lore says they eat fruits), so feathers would have been scarce, triggering competition, and becoming a status symbol.
How many vertebrae would be in the tail?
We have to consider how realistic the representation we are given are... for example consider this creature:
This creature has a single connected eye, floating eyebrows, rare proportions, an odd colored nose and a single tooth.
Let us consider how such a creature could have evolved... Wait that is a human? - You see, we have to take these representations as an approximation of how the creatures look like. That is why I have not said that the Marsupilami is not possible because a Monotreme would not have evolved a single connected eye in natural conditions.
Therefore, when we see the curvatures of the tail of a Marsupilami in these depictions, we cannot really tell how many vertebrae are there. This means that the only way to approach this question is by comparative anatomy.
The best approximation we have to the Marsupilami tail are the prehensile tails of primates. We need to notice that the number of vertebrae changes from individual to individual, and the average changes from species to species. See: Structure and Function of Platyrrhine Caudal Vertebrae
The closest data point I have is the Black-headed spider monkey that has a tail that grows 1.75 times the length of the body and has around 32 vertebrae in the tail (some individuals have 31, and some have 33).
If we extrapolate this linearly to a length of 6 times the length of the body, then we are looking at a tail of 109 vertebrae (give or take).
I may do the computation considering more data points when I have more time to look into this
And how did a cousin of the platypus and the echidna evolve a primate-like structure in body shape?
If we go by the hypothesis that these creatures separated from other Monotremes during the Jurassic, it is not odd that they would have taken a different evolutionary path. In particular, for this creature to exist, they would have convergent evolution with primates.
The influence for such evolution would be the advantage of finding fruits high on the trees and avoiding predators. At that stage, the predators would be mainly snakes, which are relevant in the South American rainforest. However, this would put them in direct competition with new world monkeys. That actually makes a good argument for them keeping a very close social structure and being territorial.
If I were creating a setting with these creatures, I would say they were problematic for a scientist, as they may attempt to classify them in Primatomorpha, considering something similar to Lemurs. In addition, that it required modern biology to compare the DNA of these creatures and settle the argument.
In fact, the only argument I have against these creatures being primates is that they had no evolutionary pressure to lay eggs, and there are no antecedents of such mutation in primates.
Jumping was not been mentioned in the question or in the linked lore, so I disregard this so far. It is hard to imagine such a slender tail to have the muscular structure for the jump.
Looking for "Marsupilami" I found the animations, how can that be possible.
Certainly, that is not possible only with the muscles in the tail. Although for the jump, what you really need is a strong force trying to extend the tail.
We engineer this if we consider the degrees of freedom of the tail. The way to do it is by having a long ligament running down the tail. Muscles in the back of the creature would then pull that ligament in such way that it extends the tail. The bone structure must be such that it locks in the backward motion. That means that the tail will not be able to curl up (it will curl down instead, so knots and spirals are still viable).
These muscular structures would have to be very strong, and be exercised often. Note that if the tail is stronger extending than curling, it makes sense that you will want to make knots for attachment.
The tail punch
I would say that hitting with the tail is just the same movement (extending the tail) but instead of doing it to propel itself, it is done in the air as an aggressive gesture.