Some, but this seems to be on a much wider scale, and geared towards assisting companies, rather than consumers. The PQIA isn't really about comparing oils, but to check oils to see they're meeting the claims they specify, particularly the API claims.
On your list, 1, probably 2, 3, 5, 6, 7, 8, and 9 really aren't within their mandate, and 4 might be, with a wording tweak, since they aren't assessing the overall quality of engine oils on the market, but are looking to get off the market things that really shouldn't be characterised as motor oils in the first place.