Are there any rules regarding construction near a private strip, such as stand off distance?
Best is to contact your FAA regional office as the details of the procedures seem to vary a bit depending on your region, and the use of the airport. 

AC70/7460 gives guidance the lighting and lighting of obstructions that may affect the airspace. The first paragraph reads:


  A sponsor proposing any type of construction or alteration of a
  structure that may affect the National Airspace System (NAS) is
  required under the provisions of Title 14 Code of Federal Regulations
  (14 CFR part 77) to notify the FAA by completing the Notice of
  Proposed Construction or Alteration form (FAA Form 7460-1). The form
  should be sent to the Obstruction Evaluation service (OES). Copies of
  FAA Form 7460-1 may be obtained from OES, Airports District Office or
  FAA Website at http://oeaaa.faa.gov.


More information can be found here   here.

The reporting requirements are laid down in CFR 14 part 77 §77.9