- I would like to know of a reference for the derivation of the quadratic order terms for the Riemann tensor, scalar curvature and Ricci tensor and for $\sqrt{-g}$?

Many introductory GR books and lectures: Carroll, Weinberg, Feynman lectures on gravitation, Misner, etc. You should work it out on your own, though. You just need to split the metric in a background plus a fluctuation and treat the fluctuation as a tiny perturbation.

- Further on such an expression how does one impose that the background is say $AdS$? (even for the linear case as linked above)

The background metric is the AdS metric instead of Minkowski. What in a flat background are the components of Minkowski metric are the components of AdS.