The A320 would have TCAS. The King Air would have a transponder. TCAS interrogates transponders. So, in flight, the A320 would "see" the King Air and the TCAS would issue traffic and resolution advisories.

Lots of GA aircraft have TCAS-like systems, where they listen to secondary radar responses from other transponders. So, when airliners or ATC interrogates planes, these other planes hear the responses. And, of course, ADS-B makes all of this easier and more precise.

But in any case, TCAS is not designed to prevent a runway conflict. It doesn't know about flight intentions: that the Airbus is going to be climbing a bunch soon.

Not just an airliner thing; my dad's four seater Grumman will start squawking at you if another plane is too close. ADS-B became a requirement in general aviation in the US on Jan 1, 2020.

Note this is just ADS-B out. In isn't required.

Yes. I would suspect that a King Air owner is fairly likely to have sprung for something like Foreflight and a Stratus, at least, but in either case the A320 is likely to have had reasonably good information on the King Air's altitude and heading.

Yup. But 8 miles out, descending in your general direction, isn't a clear signal that they're planning on landing straight in on the opposite runway from where you'll be taking off.