Interesting set of questions. I think that to answer them all, we have to lay the ground rules, though.
First, we know that the Polyjuice Potion can be used to transform one human/part-human into another human being. Since Fleur Delacour, a part Veela (not whole-human), could transform into Harry (a total human), we might be able to assume that even an animal or giant could use the Polyjuice Potion, as long as they are trying to turn into a human and not any other species. Look at what happened to Hermione in CoS with the cat hairs.
Polyjuice Potion is also pretty much like a holographic disguise from some sci-fi story. The person just takes on their appearance, really, not so much their acquired magical powers, knowledge or skills.
So, I'll try to answer your dot-point questions. Just please be aware that they are my thoughts and may be completely wrong. I will do my best, however:
1) If a person transforms (using Polyjuice Potion) into a human who has learned to become an Animagus, then, as follows from my third paragraph, the Polyjuice-user should not, theoretically, be able to transform into the hair-donor's Animagus form. Animagi are not born, they're made - with a lot of effort involved, according to Professor Lupin in PoA. Now, if someone turned into Nymphadora Tonks, who is a natural-born Metamorphmagus, using Polyjuice Potion, then we may be able to assume that the Polyjuice-user might "inherit" the same power, but would require training to master it. That's pure speculation, though.
2) This one's more tricky. An Animagus Polyjuiced into a non-Animagus should be able to transform into their animal form because it is a learned magical skill. The person underneath the Polyjuice form is still a practiced Animagus, right? So that should make it possible.
3) Another hard one. Let's use Sirius Black as a hypothetical example of an Animagus here, to avoid confusion. Also, let's say that he borrowed one of Harry's hairs for the Polyjuice Potion. Now, if the Animagus animal form overrides the Polyjuice Potion's effects, then technically, Sirius taking the Potion as a dog would not change him at all until he willingly turns back to a human. I'd say, though, that instead of looking like his usual human self, Sirius would now look exactly like Harry. So we'd see the great black dog transforming into Harry Potter. That's one theory.
Another is that the animal form of the Animagus would change into a version of the hair-donor's potential animal form. So, back to using the Sirius/Harry example. If Sirius takes Polyjuice Potion as the black dog (with Harry's hairs added), then the dog form might change to resemble something like what Harry would look like as a dog Animagus (or maybe even another animal completely, but I doubt that). So the dog might have circle markings around its eyes to mimic Harry's round glasses (like McGonagall's square spectacles on the tabby cat form) and untidy fur, like Harry's scalp hair; maybe even the green eyes. And when this black dog transforms into the human form, we'll get the perfect duplicate of a human Harry.
Let's reverse something now. What if the Polyjuice Potion overrides the Animagus animal form? Well then, presumably, the animal would be forced into turning into the hair-donor's normal (human) form. E.g. Sirius, the black dog, drinks the Potion and instantly turns into Harry Potter (the human).
This isn't a straight-forward answer, but these theories are the best I can do.
4) If the hairs (I know that the Polyjuice Potion doesn't require the body part to be hair, but its the most common, so I've been using hairs as the example throughout this answer) put into a Polyjuice Potion have been taken directly from an animal, the Polyjuice Potion user would turn into a human-animal hybrid, like Hermione does in Chamber of Secrets. So my simplest theory would be that the same would apply to the Polyjuice-user if they had taken the hair from an animal-transformed Animagus. The hair/fur/scales/feathers (etc.) taken and added to the potion would still come from an animal, technically, right?
I think the result would be either a human-animal hybrid or the Polyjuice-user being transformed into the Animagus's human form. E.g. Harry drinking Polyjuice potion with Sirius' dog fur in it and ending up with a face covered in fur, or as human-Sirius Black, respectively.
Well, that's really the best I can do without going too far into the speculative side of things (even though I've crossed the line slightly in some places).