I saw a cat killing a mouse. Could have stopped it, but didn't

It is the mouse's karma to die, and the cat's karma to kill the mouse. We recognize that animals live in the realm where they cannot comprehend and benefit from the dharma. You could have saved that mouse, maybe, but it would not change the fact that some animal would most likely kill it. We can only direct our own karma, not the karma of others. You can save a cow from slaughter only to have it killed in pasture by a wolf. Do that which is compassionate, engaging in altruistic loving kindness and generousity. Do not beat yourself up because you cannot or did not save a single creature.

If one sees a judge passing death sentence to a criminal, can one stop it? One can only pray that both the judge and the criminal will not go to the lower realms, but one is not sure, and perhaps both judge and criminal can go higher realms- one is never sure of samsara! It really depends on motivations, karma etc which one who is not enlightened will never tell...

Thus focus on one's own enlightenment for the benefit of all sentient beings, for now one cant judge activities of mouse and cats, predators and prey.

If you had small children (babies) in your home, would it be good karma to allow pests to accumulate in your home (like mice, rats, c**kroaches, fleas, ticks, gadflies, etc) that might bite your children?

I lived in a forest monastery once where (in the past) there used to be tigers & mice that chewed the dhamma books. So dogs were introduced so the tigers would attack/eat the dogs rather than the novice (child) monks and cats were introduced to eat the mice.

To make oneself paranoid or worrisome about cats & mice is bad karma but to not stop a cat eating a mouse is not bad karma (just as 'cleaning' your house of dirty pests in not bad karma).

The teachings of karma are for the purpose of keeping people free from trouble (rather than for imprisoning people).

In the strict monks rules, a monk that kills a human being for any reason is expelled from the order. However, a monk that intentionally kills an animal must only confess the action. As for allowing or wanting a cat to kill a mouse, there is no offence.

For a monk to empty a water jar with mosquito larvae in it is an offense. However, for a monk to place a cloth over a water jar to prevent mosquito larvae is not an offense. What is what here?

Next time around, try to stop the cat from killing the mouse, and then give the cat some food to eat. That'd cover all grounds: saving the mouse's life without the expense of the cat going hungry.

IMO there's a similar concern with people: outside my house I see people raising farm animals, in order to kill them for food. What should I do about that? Try to chase the people away?

I think, probably not, even if only because I think that's likely to be ineffective (i.e. that kind of tactic on my part wouldn't stop people from killing).

I suppose that the Buddhist equivalent of a "sin" is klesha or kilesa. I suppose or it feels to me as if it probably is a sin, several sins: torpor (not doing something about it), ignorance (not knowing what to do about it), conceit (thinking that you could do something about it), hatred (not liking what you see)...

Technically it's not breaking a precept: you're responsible for what you do, not for what other people do. But it's probably also right to find it questionable, if someone could have saved someone but didn't.

"Chethanaham Bikkhawe Kammam Wadami" - volition is Karma

Did you not prevent it because mice are usually an annoyance and 1 less mouse makes your life better? If so, it's bad Karma. Later you may make up an excuse like "not wanting to interfere with nature". But what matters is the intention at the time. If you were wishing for the mouse to escape, it is good Karma. If you actually saved the mouse, it's more good Karma. If you were neutral based on Upekkha(contemplating on Karma and Vipaka), it's good Karma. If you were neutral based on indifference born of ignorance, it is bad Karma.