How much force can the landing gear of a Piper Arrow withstand?
The Arrow is certificated under 14 CFR 23, and the landing gear loading limits are specified in the ground loads and landing gear sections. I don't know much about this subject and the regulations are fairly technical, but 23.473 apparently defines the load limit. I don't really understand it myself, but it looks like the landing gear has to support at least 2.67 times the aircraft's weight in a landing:


  No inertia load factor used for design purposes may be less than 2.67,
  nor may the limit ground reaction load factor be less than 2.0 at
  design maximum weight, unless these lower values will not be exceeded
  in taxiing at speeds up to takeoff speed over terrain as rough as that
  expected in service.


In reality they should be designed to withstand more than this, as there are manufacturing uncertainties which can affect a component's performance. 

Someone who's actually an engineer could probably explain it better (or correct me, as the case may be!).

As for testing, my limited understanding of the certification process is that the manufacturer has to prove to the FAA that the requirements in part 23 have been met, but there's no specific way of doing that. And indeed, 23.721(c) says:


  Compliance with the provisions of this section [i.e. the landing gear section] may be shown by
  analysis or tests, or both.