If the law is universal then it makes sense that courts can rule universally. If the law is arbitrary and every judge is basically a ruler of their own domain, which directly conflicts with the idea of Rule of Law, then it would be invalid for a lower court to make universal rulings.

It is precisely the universality and supremacy of the law which gives the lower courts the authority to make universal rulings. Just because something is federally illegal in one federal district doesn't mean it is legal in another.

The lower court doesn't have more authority than the appellate court if the appellate court can overturn the lower court.

No court has the "authority" to make arbitrary unlawful rulings.