You're lucky the drive came off! I bet that was making a heck of a racket. Salt water?
When you turn on the hose, water won't pass the impeller by much probably only a trickle till you start it, it depends on what position the blades of the impeller are in when it was last run. It won't flow through the engine till you start it. Depends on how Merc had it set up, some engines the water will go straight out the exhaust as soon as you start it and won't exit the engine, till the thermostat opens.
I've pulled the drive every season to check for water in the bellows, I've replaced the bellows twice and the gimble bearing once over 18 years worth with this boat. I find that the Cobra bellows (same as Volvo SX) can last as long as 10 years before you get little cracks in the folds. Probably a good plan to change them as soon as you see any small cracks starting. Also if in salt water, knock off any barnacles you see forming on it when you pull the boat out, they can cut the rubber.
