Typically you can run the filter with a solar cover on. Only exception for concern I can think of is if you have in-floor cleaning jets that shoot water up the wall which then ends up on top of the blanket.
No reason you can not put the safety cover on whenever you like ... other than the hassle.
Sure you can leave it on. The condition of your pool after the fact depends on many factors, but covering it should actually help maintain FC levels as the sun will not consume it and will prevent debris from entering the pool.