Probably those questions needs to be polished and stated more clearly to receive a precise answer. I'll try to add something regarding the second point (the first I'm not sure I understand): from the point of view of VNM-rationality, which is the only guarantee that an agents has a utility function, you can only deduce that utility order-type is isomorphic to R, the set of reals. So in full generality, you cannot deduce anything about the dimensionality of the utility function before stating which actually it is.
Definitely true - when I say 'yield to research' I probably actually mean 'dissolve as meaningless' in a majority of cases.
On the subject of guarantees about a utility function - is there any notion of an approximate or a revealed utility function? In the same fashion as revealed preferences in economics?
Purpose: This post is so I can record math-ish questions I have, which I want answers to, but don't merit their own post or pestering the community about directly. My expectation is they will mostly yield to a little research. If you have the answer or a relevant source, please feel free to mention.
Note : I will update this periodically, both to add new questions, mark answered questions, and to break out anything that turns particularly interesting.