I hadn't considered defining curiosity before. I looked up curiosity on wikipedia but didn't see an answer that I liked. A simple definition is that curiosity for something comes from the desire for a better mental model of that thing.

From my perspective in psychology and anthropology, evolution provides a better definition. The desire for a mental model is created mostly by the nature/nurture split in gene-culture coevolution, in which information is expensive when stored in DNA and cheap when stored in culture, hence creating a generic need for information intake, and thus a generic emotion to drive that need. "Curiosity" can then be defined as that emotion.

The circumstances which prompt that desire are determined by what "fun" is. Defining "fun" has a short and universal answer with a severe discursion into a long heuristics-and-biases approach. Judging fun requires the mind to make evaluations which are inherently difficult for System 1 to approximate, so behavior in practice very poorly approximates the short-form answer, especially as videogames explore its boundaries past what evolution is able to capably fit.

Both "curiosity" and "fun" have the problem that they are only linguistic concepts which converge as approximations to what people feel. These feelings have an unambiguous origin through evolution, which is the best way to create a description free of "common knowledge" (which relies on shared understanding to cover up vagueness). Even though the underlying definition is clear and consistent, people's usage is not so clear or consistent, and some tolerance for linguistic fuzziness is needed.

There are a few adjacent emotions which don't cleave in canonical ways, so the definition is also not canonical. I picked the emotion for mental modeling because that seems to be the origin of humans' approximate definition, and hence will best align with what people refer to in normal conversation. But you could also pick the exact concept that this emotion tries to approximate, and that would be an equally valid definition, just different. Or you could talk about information instead of modeling, but I won't.

Defining things through evolution may not be useful for laymen, because it's hard to tie evolution to behavior. But at least the definition can guide understanding even if the details are impossible to figure out. And this way, I get to sidestep your 4 questions by embedding them in the heuristics and biases that inevitably stem from an evolutionary approach.