Do all airports have the authority to impound aircraft for overdue fees?
Yes, at least if the airport is city owned (or owned by a municipality). For example Philadelphia's airports are city owned and they have landing fees as well as default repercussions 


  1.05. Payments
  
  Non-tenant Air Carriers and Operators not providing adequate surety or
  security in form and amount satisfactory to the Department of
  Commerce, to ensure payment of fees and charges, shall be required to
  pay all incurred fees and charges prior to aircraft departure.
  
  ...
  
  
  If any charges remain unpaid for a period of twenty (20) days after the due date, 
  
  the City may: 
  
  a. Bar operator from the Airport and
  the use thereof; and/or 
  
  b. Deny operator the use of Airport
  facilities; and/or 
  
  c. Commence such actions at law or in equity as are
  deemed in the best interests of the City.
  


I see no reason impound would not be covered under section C there. Privately owned airports are a different situation but I imagine they have some kind of legal recourse.