Summary: the problem with Pascal's Mugging arguments is that, intuitively, some probabilities are just too small to care about. There might be a principled reason for ignoring some probabilities, namely that they violate an implicit assumption behind expected utility theory. This suggests a possible approach for formally defining a "probability small enough to ignore", though there's still a bit of arbitrariness in it.
Only in the deterministic case. If you have uncertainty, this doesn't apply anymore: utility is invariant to positive affine transforms, not to arbitrary monotone transforms.
Any risk-neutral (or risk-seeking) preference in any quantity.
I am not sure I understand. Uncertainty in what? Plus, if you are going beyond the VNM Theorem, what is the utility function we're talking about, anyway?