If a pilot were to call mayday and say they're landing on runway 6 everyone would be trying to ensure runway 6 was clear. The only way 6 would be blocked is if there was something already on it that couldn't be moved in time. Think of a mayday call as the lights and sirens of the sky.

Pretty sure the system doesn't parse ATC audio but as other posters have noted can use machine readable METARs and other data to get a sense for weather and available runways. (Note that the system does NOT parse NOTAMs yet per https://skiesmag.com/features/virtual-co-pilot-garmin-autola...) Beyond that am guessing that it declares its intentions and hopes ATC can pave a clear path. They are generally exceptionally good at doing this for aircraft in distress; just squawk 7700, state your intentions on 121.5, they will make it happen - you're allowed to do literally anything and violate every rule in the book if it's required for safety of flight. You can land on an active military base or a major commercial airport runway if you need to. Mind you, there may be some paperwork to fill out on the ground after, but while you're in the pickle everyone is there to help.

Any reasonably sophisticated jet (of which this is clearly one) will have access to more or less realtime weather data, as well as NOTAMs (notices to airmen) regarding closed runways and such.

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.

One way. The purpose is so that other pilots are aware that there's a problem. In a lot of US airspace there is no air traffic control, and most strips do not have anything resembling a control tower.