I am not a biologist so my answer will not be accurate and make some sweeping generalizations, but I want to play mad scientist here and give you some ideas to ponder
A four legged animal with a spine that does not generate its own body heat, instead gaining it from the surrounding environment. This leads to a low metabolic rate. The young generally undergo metamorphosis from larva living in water with gills to an adult air-breathing form with lungs.
An animal with a spine that generates its own body heat, this leads to a high metabolic rate. It also possess a neocortex (a region of the brain), hair, three middle ear bones and mammary glands
a slender, flexible limb or appendage in an animal, especially around the mouth of an invertebrate, used for grasping or moving about, or bearing sense organs.
Probably not but as outlined in Ghotir's answer the Platypus exists so lets assume on an alien world that your Bunopus is the million to one shot and comes about. I think it would have to happen on a low gravity planet which would make the tentacle viable as an effective form of locomotion. Octopus do 'walk' on the sea floor (but aren't very good on land)
So the Bunopus needed a reason to come onto dry land. This could be because the water it lives in had low oxygen levels, or possibly went through periods where it dried up. This would lead to evolving lungs to come to the surface and breath, or survive during the periods of drought.
They will then need a reason to stay on land. A good food source, perhaps plants or smaller invertebrates. From my quick research I find that there shouldn't be any predators that will pick them off in this early stage. They will need to be at the top of the food chain.
Amphibians have primitive lungs that are rather inefficient, and so take in a lot of oxygen through their skin. To do this they must stay moist, so the fur could be a system used to trap moister and protect them from dying out. It might be fur as we know it, but elongated scales developing into a feather like covering. Maybe without the center spine but just the barbs resembling fur.
Given that a definition of amphibian is four legged (tetra-pod) I would say that four of the tentacles evolved for locomotion. The remaining two are sensory organs, identifying food; detecting electrical signals in the water; etc or feeders. This could come about from needing a way to catch faster prey, or by needing to lift the body higher to reach foliage on tall plants, and the remaining two from an advantage provided by what ever you choose from them.
I would imagine that the Bunopus' sight is quite bad, or non-existent. They would then need a form of threat or prey detection. This could lead to an audio sense, and over time develop the three-boned structure of the ear found in mammals. Over time these could become bigger and more sensitive providing a better survival/catch rate and making for healthier mates.
I think I have covered the basics of the traits you asked. I know I have left a lot out like why they would feed their young and develop mammary glands. Also I think I covered why having these traits would be useful but not how they came about from small genetic mutations. But hopefully this might point you in the direction you want to head and really get the mental juices flowing.