I'm curious about a point of law. If an appeals court in another district makes a ruling opposite to or incompatible with the Dover ruling, what happens to the Dover ruling?
Strictly speaking, the Dover ruling applies only in the federal 3rd Circuit (which is the federal circuit PA is in). Courts in other circuits don't overrule it even if they decide differently, because they don't have jurisdiction in the 3rd Circuit.
In practice, other jurisdictions will probably look to the 3rd Circuit decision for nonbinding guidance. If another circuit decides a similar case differently, that won't create an actual conflict but it could reduce the practical amount of "reliance" that other courts put on the Dover decision.
Can the Dover ruling be overturned by another court, and if so, can this then be appealed?
The Dover ruling can be overturned only by a higher federal court (an appellate court, which would hear the case only if there were grounds for an appeal). The appellate ruling could itself be further appealed to the Supreme Court, which would decide whether to "grant cert" (agree to hear the case by issuing a writ of certiorari).
By the way, you asked, "If an appeals court in another district makes a ruling . . . " Such a ruling would come first from another (trial-level) federal circuit court, not from an appellate court. Of course the appellate courts could get involved later, in any cases where there were grounds for appeal.