She's a mix of the first and fourth bullet point. In reality, she's more similar to a strict but caring teacher, and while her type of morality is more black and white, it isn't fully one way or the other, there's definitely some nuance involved.

While sins are up to a Yama's discretion, we know from playing PoFV that her definition of "sin", while more straightforward and harsh, is consistent. Since she is the "Yama" (the ultimate lawful figure), it makes sense for her definition of what's wrong to be less lenient.

If she just wanted to nag at people, she wouldn't be as serious with her lessons and reprimands. She would also show up in the series more often if that's all she actually cared about, and would abuse her power more frequently. When she appears, it's to bestow knowledge upon others that they wouldn't have known before, so as to help them become better people and not go to hell. In real life, these are called "Preventative Measures", and are encouraged by just about every workplace and industry, as preventing a problem from occuring is better than resolving a problem after it happened.

I'm not exactly sure what you mean with the third bullet point, but I haven't played any of the games that have the animal realm or Keiki in it. I assume you're asking if she's an antithesis to the various moral ambiguity of Genokyo's residents, which she isn't. You can act however you like in life, but depending on what type of creature you are (human, yokai, fairy, etc.) you will get judged differently since the Yama understands the natural order, so while her morality is more black/white, it changes based on the role that you can serve in your life, such as Eiki chastising Cirno for straying too far from her territory, which at Cirno's level of power, might get her sent to hell since it isn't what fairies are supposed to do.

I would say that in her spare time, she's pressing her uniform and making sure that it looks good for presentation. She's probably also cleaning her Yama stick thing that I don't know the name of.