It is usual for the leader of the party to remain as PM while a new leader is elected
Why I think he'll try and stop but do seem to be many more voices saying no he can't and as they've just said on BBC news PM is an appointment by the Queen - and I can't see the place thanking them for involving them.
As for how can minster work with someone they think it untrustworthy - their politicians and they'll spin it if they need to but it does look like Labour might be willing to call a vote of no confidence now so I do think it's possible he may be gone quicker than thought earlier this morning.