Who is responsible if my neighbor installed a fence that damaged underground pipe

You have two potential claims one is for simple damages that might have been caused by the installation of the PVC fence (or gate post). The other might be for personal injury due to the alleged conditions in your basement. It would be much harder to prove the second since that would require medical experts testifying that respiratory systems were due to something like mold or other environmental condition and that such illness was directly related to whatever situation was created by the damaged underground pipe. Proving the property damage might be easier but you would also benefit from having a contractor or civil engineer or other expert examine the situation you mention about the PVC fence and the gate post. I am not sure about the gate post because it appears that is something you did, not your neighbor. I think you would first need to prove that the fence damaged the underground pipe, and again that means having someone who is expert in that field who can prepare a report that could demonstrate that is the case. Obviously, I think having an attorney assist you would be of help as this is not a matter I think that a layperson can handle.