when someone falsely accused you for anything could you go to jail?

It is highly unlikely but not impossible. The accusation is meaningless unless the police and district attorney have adaquate evidence to convince a judge to convict the person in court.

People go to prison for murder they weren’t in the same country for so yeah anything is possible. Don’t talk just ask for an attorney if it’s serious

Yes.

I have a neighbor that threatened to call the police saying I abuse animals (which I don't) You cannot be punished without a bit of evidence first. If a detective talks to you (or a cop even) make it clear that you're innocent and be aware that some asshole police will ask questions in a way that would make you seem guilty (teens are interrogated like suspects of anything in my city- if a teen is walking at midnight, police will assume automatically they're on drugs or runaways and they'll even ask them questions to make their answers seem a certain way. In a nutshell, accusations you can disregard as long as you look out for people trying to get you to say things they could use against you. Hope this helps!

In theory, yes. But they don't seem to be doing that much.