New Quaker State Oil w/ Zinc
Are your giving me used oil test places?
Yes.
If so, why would you do that in a thread on new oils?
For a couple reasons, the most important because it's something the average Joe can afford. You can send a sample of new oil off to Blackstone or Wix, and the report you get back will give you a basic but thorough list of the oil's contents, limited by the ability of the testing methods. It will tell you how much zinc, phosphorus, magnesium, boron, molybdenum, calcium, etc. With that information, you can get a basic understanding of what is in the oil in terms of anti-wear compounds and detergents. Basic viscosity tests are also often part of the service.
No worries since it was pretty much an empty answer. You can finish it up with where to get used oil tested for if possible. NAPA is the only place I heard of on your list.
Sorry to disappoint you. Google any of the names I gave you and "used oil analysis" and you'll have your answer how to use their services, and I suspect several more.
Who else tests “NEW OIL?”
Not used oil.
For that information you'd want to Google something like "labs that perform ASTM tests which determine SAE, API and ILSAC standards." Those are performance tests that determine grades and ratings which you'll see on oil containers. Expect to see a webpage with contact info. You tell them what you want, and they'll send you a quote.
Those tests can be very expensive. Some involve running engines for many hours, tearing down, and measuring wear. They are not tests used by consumers, as they can cost in the tens of thousands of dollars to be performed by a certified lab. Those tests are so expensive that the cost is a reason, not necessarily the only reason, that some of the smaller blenders don't have API certification on some of their products. Some Amsoil, HPL, Red Line, Driven products are right at the top of the heap in terms of quality but aren't certified. For one thing, their targeted customer doesn't care about certification. But another reason is that the cost of certification has to be recouped in the price of the product, which would make the price of already expensive oils beyond reason.
So, with a few exceptions, new oil testing is really something only done within the industry as a necessity to assure consumers products meet standards required for their vehicles. The PQIA, referred to above, does some nice testing, which is a great benefit to consumers. But they only do a few of the more basic, inexpensive tests, and not the full battery required for certifications.