Yes, it will be water flowing to the shaft log seal area when the engine is running. Even if this flow is interrupted and the passage clogged, it will be water in the area around the seal/stuffing box but it will be quite stagnant since there is no flow of a fresh supply of water. I suggest you remove the hose and fitting at the shaft log at the next haul out and inspect it for any obstructions.
You can disconnect the hose while running, but expect a stream of water coming in when doing so (start the engine and do this at the dock). It will not be a huge amount since the opening is rather small and you can easily put your finger over the opening to stop it. You should have a steady stream of raw water coming from your hose while the engine is running.
I suggest to do a search in the older posts regarding fuel tank removals, it's several references since this has come up frequently over the years and has been covered from all angles. In short, arm yourself with a couple of dozens of carbide blades and a saws-all and plan on spending the next couple of weekends cutting them up. Then have someone build smaller stackable tanks, also covered in earlier posts.