The fact that airplanes that are low on fuel have absolute emergency priority has occasionally resulted in, uh, incidents: https://www.youtube.com/watch?v=D3RUWvRRGLQ
I'm not familiar with the system's details, but I would assume that it would pick the nearest towered airport with emergency services that met the requirements of the landing performance. Landing autonomously on a strip in the middle of nowhere, with nobody around, isn't likely to be particularly useful in an emergency unless the aircraft is no longer airworthy - and even then, if you can keep it in the air, going somewhere big with emergency services is a better idea.
A towered airport will have air traffic control (not all small strips do), and if the jet sets the transponder code to indicate an emergency, and is making blind radio calls about intent as to what runway it's going to land on, ATC will then clear the airport for the incoming emergency, and keep everyone else out of the way - and roll the trucks to meet it once it stops. I doubt it will bother getting off the runway autonomously - just come to a safe stop, shut down, and then it's someone else's problem to taxi it off.
Aviation is somewhat nice in that way. Emergency traffic has priority. If I were to be flying a Cessna 152 along Lake Michigan and had an emergency with O'Hare being the nearest airport, I've got priority, and the oceanic heavies will get out of the way until it's resolved. Now, you may have some serious questions about what the problem was, but this is the sort of stuff resolved after the fact, on the ground.
There will be questions, but there is a very strong doctrine of "A pilot in an emergency can do whatever the fuck they think is best", to the point where they are explicitly trained to ignore ATC orders in an emergency, if they don't agree.
A pilot would have to do something extremely malicious before the FAA would even start to question the choice of emergency landing airport. They don't want pilots second guessing their decisions out of fair of reprisals.
Mayday call is used on life threatening situations.