unless you have water lift mufflers or improperly designed exhaust I don't see any risk of running the water while the engine is not running.
My concern has always been that the city water flow is much less than the raw water pump flow and I wonder if it provides enough lubrication for those big impellers.
I would turn on the water before closing the seacock. closing the seacock would be necessary because so much more water comes from the outside that the fresh water would be diluted.
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