The FAA actually does keep engineers on staff, and the aircraft manufacturer will have to explain and demonstrate how their aircraft complies with thousands of different requirements. This is actually one of the larger sub-controversies of the whole MCAS issue; the FAA has been progressively delegating more and more of airworthiness compliance to Boeing's engineers and trusting their results instead of verifying it themselves.

With regards to the regulations, they're excruciatingly detailed. I encourage you to read 14 CFR 23, Airworthiness Standards: Normal Category Airplanes, and 14 CFR 25, Airworthiness Standards, Transport Category Airplanes.