is false advertising a civil or small clams case?

You are mixing apples and oranges.  Civil cases are cases brought by aggrieved parties to redress private wrongs, as opposed to criminal cases which are brought by the government to address crimes.  All small claims cases are civil cases. Whether a case belongs in small claims court, as opposed to another court handling civil matters, depends on the amount and type of relief sought by the plaintiff.  If the plaintiff seeks $500 damages and no other relief, he/she/it should sue in small claims court; if he/she/it seeks $100,000 damages, the case cannot be brought in small claims court, and must be brought in a court having jurisdiction over more substantial matters.   Also, in many jurisdictions, small claims courts do not have jurisdiction to afford relief other than damages, such as an injunction so that, regardless of the money involved, you may have to bring that case in a court of general jurisdiction, not small claims court.  I don't know that the jurisdictional limit on small claims court is in Ohio, but you should be able to find out on the Court's website, or by asking the clerk of the court.  There is also a question of whether you want to sue in Federal or State Court.  If your case presents a federal question, for example if at least one of your causes of action is based on the alleged violation of a Federal statute (depending on the particular facts, false advertising may be a violation of the Lanham Act, a Federal statute), you may have the choice of suing in either Federal or State Court.  Sometimes, you won't have a choice, as Federal Courts have exclusive jurisdiction over claims brought under certain Federal statutes, for example cases brought under the Patent Act.