Hi and Welcome!
Generally, pools are NOT emptied for the winter, but are merely winterized to protect the plumbing and mechanisms for freezing and damage.
I cannot answer on pricing on closing because:
a) I'm the only one who has ever closed MY pool.
b) My pool is vinyl-lined
c) Different areas have different pricing depending on the local market.
While it SOUNDS expensive to me, whoever you get to close it should let you see how it's done and then you'll be able to do it.
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