Can a Federal Court here an Appeal of a State Case?

Generally, no. The only federal court that can hear appeals of state court cases is the US Supreme Court.

Can a State Appellate Court Hear an Appeal of a Federal Case?

No. This is not a possibility.

Can Federal Appellate Courts hear matters of state law? 

Federal trial court decisions are appealed to the Federal Circuit Court of Appeals or via special writ to the US Supreme Court.

Federal District Courts and Courts of Appeal cannot review decisions from state court cases.

Also, state trial or appellate courts can never undertake appellate review of decisions from federal court cases.

State trial court decisions are appealed to the state intermediate court of appeals or the state’s supreme court.

There is, however, one exception to this rule.

The US Supreme court may review the decisions of state supreme courts. If the court’s decision appears to conflict with federal law, (such as a statute, treaty, or the US Constitution).

US Supreme Court review of state supreme court decisions is most common when the state court upholds a state law that could potentially violate the appellant’s constitutional rights.

In such a case, the US Supreme Court may issue a writ of certiorari or accept a request for an appeal of the state Supreme Court’s decision by the losing party.