That’s called arbitration. As long as everyone opts in, you’re good to go

If you do so on property that you actually own, it’s not government

What if it’s property we commonly own and decide its use through a system of elected officials?

Let’s say we have a small state established and someone moves to our state in a manner what breaks our state laws. Can we remove them?

As an anarchist, as you long as you respect private property rights, I cannot object

DIY

So they move to a piece of land owned by you?Yes you may remove them. The land is yours. If they resist you may remove them by force. Force should be appropriate to the situation. If telling them to fuck off from your lands is enough then you should do just that.