The potential problem with fishing line is that it will cut or damage the prop shaft seal and cause either loss of oil or water to get into the lower unit oil. So, you'll want to check it a couple times by pulling the lower unit drain plug and letting about a teaspoon of fluid out. If it is clear, you're fine. Milky means water. The fore and aft play in the prop shaft is normal. Spinning while out of gear is not a cause for concern.
You have not said what size or brand of engine, but the prop normally engages the shaft with splines, so the nut being loose or the cotter pin missing should not have any effect on if it spins. If it moves the boat in gear. the splines are engaged.
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