It's unclear from the initial term-seeking and this response, so just to make sure: it's a standard term, the usual reference is

• M Li, P Vitányi, An Introduction to Kolmogorov Complexity and Its Applications.

"an unknown thing that is a probabilty distribution" would be a thing that fullfills the definition of a distribution without knowing much else about it. If you have multiple dice and you get a number but don't know from which dice it came from (say d6, d20 and d4) the fact that is coming from a die could affect things.

"A thing that a probability distribution doesn't exist for" where the birth of the value is so nebolous there are no good boundary conditions what it could have been. https://xkcd.com/221/ if there was no "//chosen by fair dice roll" one woudl have no idea about the selection process. This migth potentially be too shaky a concept to be interesting. One could have the opinion that single cases always have a reference class even if they belong to multiple such ones and a single "right one" doesn't exist. But what happens if you single case is so bizarre you can't describe it? What happens if you single case is the whole of all the reference classes it belongs to (Saying that "X is a smerp" is not helpful if the whole definition of a smerp is to point at X and say "like that")?

Other failure modes could be to fail to have properties of probabilty distributions. Negative numbers, imaginary amounts? Not an unknown probability distribuiton because its not a probabilty distribution (probably is still very unknown). Questions involving infidesimals like "Given a uniformly random number between 0 and 1 is it twice more likely it to be in the interval from epsilon to 3*epsilon than in the interval from 3*epsilon to 4*epsilon" The probability math might have trouble because measures make limited sense rather than the entities involved being numbers. Are transfinite ordinals numbers? If your random pull is aleph-null does it have any sensible distributions to be drawn from? Dealing with other "exotic" numbers systematically might get tricksy.

Sure! A Q-Q plot allows you to see how well a collection of data points fits a particular probability distribution. For example, let’s say you generate 20,000 random points from a student T distribution, and try to fit them to a normal distribution with an identical mean. The Q-Q plot may show that your points diverge from the distribution expected if the data was drawn from a normal distribution, but that it is a good fit for a student T distribution. I’ve found (in my limited coursework experience) that you often need lots of data points to get clarity on this, because the differences are most apparent in the extreme values far out in the tails.

This is connected to a broader class of mathematical challenges, such as curve fitting. If you look up courses on predictive analytics, you’ll find many examples of challenges related to the one you articulate here, along with tools for solving them!