Is it possible for states to make laws saying that cop-likes can't lie about things? And then for the state to sue the federal government if they violated that?

Because the truth is that requiring a warrant won't do the trick. If the cop-like says he doesn't need a warrant I'm going to listen to him. I should always comply in the moment or "we'll do things the hard way". Then afterwards maybe I can sue.

No, it couldn't affect ICE since the federal government has sovereign immunity and all federal officials enjoy qualified immunity in the course of conducting federal business.