A: That depends on your municipality. However, most—if not all—require a permit to ensure compliance with local code requirements for a fence. I recommend contacting your county or city building department to confirm the specific requirements before starting your project.
A: The standard practice is that any lien against the property should be paid off at or before closing. You should also have obtained title insurance, which generally covers liens that were not satisfied. You should have received a copy of your title insurance policy sometime after closing, and the lien coverage (or any exception) should appear on the policy.
At this point, you should contact an attorney to review all of your closing documents and advise you on the best course of action.