1) Exogenous controls means that
E(error|x) = 0. In the context of an IV regression it means that the exclusion restriction is satisfied or
Cov(z,error) = 0. To quote Wooldridge, "In the context of omitted variables, instrument exogeneity means that z should have no partial effect on y (after x and omitted variables have been controlled for), and z should be uncorrelated with the omitted variables."
2) Adding (relevant) controls is always the efficient thing to do because you don't want to be throwing away information. If you have more than one instrument, the linear combination of the instruments is the one that is going to be most "relevant".
As for testing whether the instrument is valid, one can do a couple of things (like a test for over-identification) but it doesn't completely guard against all criticism, especially the one you point out. Whether an instrument is good/valid, comes from theory and it is possible even within the social science to find such instruments. For examples and further discussion please refer to Wooldridge, J. (2015). Introductory econometrics: A modern approach. Nelson Education and Angrist, J. D., & Pischke, J. S. (2008). Mostly harmless econometrics: An empiricist's companion. Princeton university press