If BoJo did resign, who'd get the job?

[Miljah]

I should know this, but who would be 'caretaker' PM? And would it all trigger a GE?

Teresa May carried on until BoJo started. They don’t have to go immediately. Also they often have a Deputy PM or designated Minister for that role if the PM is away for example. For TM it was David Lidington. Not sure with BoJo but there must be someone. Andrea Leadsom maybe?

Shall we just give all Tory MPs a turn in the hope that one of them does something constructive?

There is no rule. Raab is First Secretary of State and thus the unofficial 'deputy prime minister,' not that there is such a thing, but in practice it would require someone in the cabinet to budge up and sit in the big chair and either attempt to command a majority in parliament or seek a dissolution, which thanks to Cameron & Clegg is no easy matter as it ought to be.