Which immune assay is best to measure inflammatory responses in general?

Several. The more "generic" tests to mesure inflammation are effected by many things. Some of the tests that doctors use for this include: ESR "sed rate"; CRP (c-reactive protein). Both of these can be false effected by various reasons, but all of them also give you an idea of inflammation. None of these tests specifically test the function of the immune system, or screen for immune deficiency.