We keep our passports in the safe in our cabin, can they get into these without the combination?
Yes, definitely. Security will be able to access the safe (as I found out on my last cruise, when I forgot my code to open the safe )
When disembarking the ship, passengers will be given an 'all aboard' time, which is usually at least 30 minutes before sail away time. Even then, if there are late comers, the ship may wait a bit longer, perhaps another 30 minutes, then the captain will decide whether to leave without the late comers. Their passports will be left with the port agents.
After a close call once due to a lorry fire causing traffic gridlock, I only ever do excursions on my own if they are under 5 miles from the port - I know at the very worst I could jog back to the port if it looked like we were going to be late - I mean, I wouldn't want to, but I could if I absolutely had to. If going further afield, I now prefer to do a ship organised tour, as they will either wait for you if there are problems, or pay for you to get to the next stop.