What specifications/parameters/tests are your customers requiring that you meet?  What about building code applications?

Standards and specifications are usually available and written for specific applications, so you need to know which ones apply to your applications.  For instance, if your material is going to be used to support loads in bending (dead loads, live loads, wind loads, etc.) then your product would need to meet the typical parameters that show compliance for that application.  You would then find an ASTM or UL standard that addresses minimum tensile strength, minimum shear strength, etc.

We don't actually supply a complete system. We may supply a piece of a system. For example, skylight manufacturers can use our product for the glazing. So we are trying to determine what test would be beneficial to us. I am going through the listing on the ASTM website regarding plastic testing, that has helped a little.

So to answer your question, these tests don't pertain directly to any of our customer requests. We are trying to cover the bases prior to getting a customer request.

Common types of tests: strength, impact toughness, UV stability (effect on all properties), scratch resistance, distortion temperature.  I am sure that this list isn't complete.
I would suggest that you look at all ASTM specs that cover the material that you are making as well as the testing specs.

For example there are probably "F" specs that cover the use of your material for glazing.  This will lead you to "D" test methods and "G" environmental degredation tests.

Some of these are tests that you will do once, and only repeat when you change process or material source.  Other tests may be done on each production lot to verify properties.

It would be worth joining ASTM.  You get on 'free' book with your membership making it a money saving deal.  You don't have to attend the meetings, but you can learn a lot and meet other poeple in the industry.

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One of the keyword that you have in your second commentary is glazing, so check your ASTM listing for appropriate tests on plastic glazing.   Also I am sure that the USAF has MIL Specs comparable to ASTM for their jet fighter canopies which I think is of Lexan material.

Find data sheets for competitor materials and see what properties and associated test methods are listed.