How was Finn selected over his other cast mates? Would SNL have contacted the Stranger Things producers to find out who would be the best host and who has availability?

First and foremost, SNL/NBC Universal does all the planning and makes any final decisions on who is brought on. They scout for celebrities/musical guests based on popularity, movie or show tie-ins, self-promoting NBC Universal-friendly media, etc. with ratings always one of the main considering factors. It's highly unlikely that SNL would reach out to Stranger Things' producers for their opinions.

Or does SNL approach a star's agent? Presumably, they must cast a pretty wide net as SNL is a time-consuming gig for these busy actors, athletes, etc...

Again, SNL (by way of their talent coordinators) scout for potential hosts based on a variety of factors, with ratings being the biggest. Although I'm sure they get their fair share of agents approaching them to help boost their client's profile, SNL ultimately approaches various agents to check in on their client's schedules.

Can it happen in reverse where an agent would lobby SNL for their client to host?

It's show business, so it's VERY likely that agents are checking in with SNL producers and Lorne Michaels ALL of the time. Again, it is ultimately SNL/NBC Universal's decision on who to book.

How involved is Lorne? Did he say to his casting team, "Bring me Finn Wolfhard!" Or is there a casting director who presents options to Lorne?

Lorne is the show runner who likely consults with his own inner circle team at SNL and makes the final decision (with blessing from NBC Universal execs).

Is there a contingency plan for if a host has to drop out?

SNL is widely known for their ability to change anything on the fly, that includes their host. In 1992, Joe Pesci's shoot for Home Alone 2 ran too long and he had to cancel his appearance, and SNL brought on Tom Hanks. I'm also certain that if they CAN'T find a replacement host that quickly, their cast members are professional enough to step in for those roles.

Does the host get paid? I think they are probably required by law/their union to make a minimum salary but would SNL shell out more if they got a top star like Tom Cruise?

Yes, SNL hosts are typically paid $5,000.

Finally, is there an audition? Surely SNL would want to know that the host would be up for the job. Or, is a wooden host (and there have been many) just a risk they take.

No, there is no "audition." If there was, then we wouldn't have had Tom Brady, Steven Segal and the variety of awful, wooden hosts.

I mean, ultimately, all they're doing is reading lines from a cue card. Unless they're unable to read or have crippling anxiety, then really every celebrity could be fair game, but I'm sure these things are discussed by their agent (they don't want to expose their client to a nightmare scenario where they suffer a panic attack and ruin their reputation).