Does the federal government have the authority to file criminal actions in state courts?
No. Federal prosecutors may only bring prosecutions under federal law in federal courts.
Federal prosecutions may not bring state law or federal law prosecutions in state court (although the federal government can and does sometimes consent to participate in a civil lawsuit in state court, most often in a probate case or a case adjudicating property rights where the federal government's only interest is a lien on the property).
As another answer notes, the authority limiting federal law criminal prosecutions to federal court is 18 U.S. Code 3231.
Part of the reasons that federal prosecutors can't bring state law prosecutions is that state law generally does not authorize them to do so. There is also a general principle in the law that one sovereign may not prosecute another sovereign's criminal or tax laws, which also applies internationally. To the best of my knowledge that is not codified, but I might be mistaken.
A state law prosecution of a federal official in state court, however, may be removed to federal court, to be tried by state prosecutors in federal court. 28 U.S.C. § 1455.
No. Under 18 U.S. Code 3231, federal district courts have exclusive jurisdiction to handle offenses under the federal criminal code.