Do I have to pay a "claim for damage" for a utility running under my backyard that I did not know was there?

Dear Sir or Madame,   This easement would have been revealed by a survey or examination of a recorded  plat (if a subdivision) in the county public records. Mere existence of an utility easement is not a title defect. Regardless, I am certain the easement was in fact recorded in the public record and was probably noted in your title commitment (assuming you got title insurance) . I can't explain why you didn't know about the easement but any reference of this easement in the public record is considered notice to you. Again, this easement would have shown up if you'd gotten a survey. If you don't get a survey , title insurance will exclude all matters (encroachment, set-back violations, etc) that might have been shown on a survey. Remember title companys are not lawyers. They can't give legal advice and do not represent you for such matters.