What are the aircraft requirements to legally offer commercial sightseeing flights in a C-172?
It needs to be airworthy, as per 91.403, which is defined to include compliance with all ADs:


  (c) No person may operate an aircraft for which a manufacturer's
  maintenance manual or instructions for continued airworthiness has
  been issued that contains an airworthiness limitations section unless
  the mandatory replacement times, inspection intervals, and related
  procedures specified in that section or alternative inspection
  intervals and related procedures set forth in an operations
  specification approved by the Administrator under part 121 or 135 of
  this chapter or in accordance with an inspection program approved
  under § 91.409(e) have been complied with.


It needs a 100-hour, as per 91.409:


  (b) Except as provided in paragraph (c) of this section, no person may
  operate an aircraft carrying any person (other than a crewmember) for
  hire ... unless within the preceding 100
  hours of time in service the aircraft has received an annual or
  100-hour inspection and been approved for return to service in
  accordance with part 43 of this chapter or has received an inspection
  for the issuance of an airworthiness certificate in accordance with
  part 21 of this chapter. The 100-hour limitation may be exceeded by
  not more than 10 hours while en route to reach a place where the
  inspection can be done. The excess time used to reach a place where
  the inspection can be done must be included in computing the next 100
  hours of time in service.
  (source: 14 CFR Part 91.409). 


Part 43 Appendix D specifies the items which need to be checked in the 100-hour inspection.

And it needs to have all the other necessary mechanic's inspections, including ELT, transponder/pitot static system, etc., and pilot checks, like VOR accuracy.