When you are using a giant harpoon to fish,and that harpoon is bolted to the ship, yes, you can leave it. As far as it swinging wildly, I doubt it. Once the line is taught there will be little give. Your ship will be pulled, or pull theirs, and the gun may swing if they decide to change direction, but for the most part the strain will be on the line and the bolts. Maybe there can be a bit of damage penalty for leaving the gun in attended, but I would expect it to be reinforced, since the builders specifically planned for it to be attached to enemy ships.
Also, yes, to reel someone in, you would need to be on the gun, but once the line is hooked on the enemy you can leave it. If you do not plan on reeling them in there is no reason to be on that gun anymore. This would allow for a deadly pursuit, depending on which ship you are flying. I imagine most dangerous would be a galleon, with the light gun on the side as a harpoon. Top engie can fire harpoon, then get back to repairs, allowing the bottom engie and gunner to open up with heavy weapons.