It's not just interaction – the actual text from the SRD is
Creatures encountering an illusion usually do not receive saving throws to recognize it as illusory until they study it carefully or interact with it in some fashion.
Let's break this down to determine its meaning. "Studying carefully" is the minimum effort required to gain a save to disbelieve at a distance. This careful study is more than any other type of remote interaction, therefore all that leaves for static "interaction" to mean is touch-range interactions. (If forms of static distant interaction less than careful study was sufficient, they wouldn't bother mentioning careful study.)
This is supported by the dictionary definition of "interact", which is "to act upon each other" – you have to act upon (and thus have the chance to note the effects of your action) a thing and have it act (or react) back in order to have an interaction.
That just covers observation, though: study it carefully, or get in contact with it. Further than that, the dictionary definition of "interact" makes that SRD text convey what dynamic situation is required: acting upon it or being acted upon by it. So, having a social interaction would qualify too: the wolf growling at you, and you standing your ground or threatening it back, would give you enough subtle cues to observe to have a chance to notice that it's behaving unbelievably.
So understanding that short text answers all of your bulleted example scenarios:
the enemy sees, hears or otherwise perceives the wolf?
No save. This is less than "careful study" and involves no interaction. Illusions would be rather fragile and useless if this qualified.
the enemy carefully studies the wolf (using a standard or other kind of action)?
Yes. This is explicitly denoted in the rules.
a PC threatens to use the wolf while intimidating the enemy?
No. But notice that this turns on who is doing the intimidating: the PC threatens to use the wolf is a social interaction between the enemy and the PC, not the enemy and the wolf – the wolf isn't doing anything to qualify as interaction. Furthermore, you're too busy being intimidated (or resisting intimidation) to be engaged in careful study.
If you mean that the wolf is doing the intimidating, then that's interaction and you do get a save.
a PC stages some sort of scene involving the wolf while bluffing against the enemy?
Maybe. It depends on what the wolf is doing, and "involving the wolf" is too vague to say. If there's any doubt about whether an interaction with the wolf is happening, no save.
the enemy (or some non-hostile NPC) touches, speaks with, etc. the wolf without there being any attacks?
Yes. This is denoted by "interact with it in some fashion".
As for saving throw modifiers for visibility, the rules make no provision for this.
I'd be reluctant to bother houseruling it either. In the case where there's no light, you simply can't use your sight to perceive an illusion, and you can't disbelieve something you're not experiencing in the first place. For illusions with physicality it wouldn't make any sense to apply visibility penalties non-visual ways of studying and interacting in the dark. As for dim or other visibility-reducing situations, I'd be hesitant to bother making up modifiers – it's just as likely that the illusion is given away by being improperly matched to the local lighting/shadows as anything else, which suggests a bonus – so it all comes out in the wash.
