Hard to imagine Britain joining both the Euro and the Schengen zones. I just don’t see the British public not thinking they’re special and demanding exceptions and carve outs just for them.
I suspect they could maintain an opt out for the Schengen, purely because a land border exists with Ireland, which isn’t part of the Schengen area either. (That being said I’d like it if Ireland joined Schengen, as it makes it just a tad more convenient when flying to the rest of the EU) Doubtful that they could avoid a commitment to joining the eurozone though.
Airports would still require travel documentation (e.g id card or passport)