It sounds like you appeared at the trial. Your remedy WAS to appeal to the county court. If the appeal deadline has passed (5 days after Judgment) there is nothing you can do. Your claims/causes of action were most likely compulsory counter claims and had to be brought in that action. You should consult with a local landlord tenant attorney to verify as I do not know all of the facts and timelines concerning your specific case.
Thank you for your question. An eviction does not necessarily go on "your record." It sounds as though a judgment was entered in the Justice of the Peace Court. If the judgment included an award of monetary damages, the plaintiff could file an abstract of judgment with the County Clerk, thereby creating a public record that may be discovered during a credit search. Generally, people do not abstract a judgment for possession, although it is possible. As for the records kept by the Justice of the Peace, you will not be able to change or alter those records if you did not appeal the decision within five days of the judgment.