What follow-up actions, if any, does the FAA take when a pilot declares an emergency in flight?
I don't know the FAA policy, but normally, you don't report deviations from the regs, unless asked to.  I have declared emergencies, gotten priority handing in the DCA area, and a variety of eyebrow raising events over the years.  However, the FAA has never asked me to submit a written report. Once or twice, I have gotten a call from an operations inspector, and answered their questions, and that was all that became of it.

In general, if you had a bad day, don't tell about it. Just get on the ground safely, and be polite to people, and hope that you don't have any further review.  Of course, you must comply with NTSB 830, but if you are required to do that, you have had a really bad day, and might want to get some help (aviation attorney).