You should have such line item in the lease about when landlord can access the house with advanced notice. If there is none, your should modify it in the future.
Their requirement is reasonable. Just work with them to get mold inspector in and make sure there is no mold (I doubt there is). Get the document and photos. You don't need to repair the floor now. (They didn't ask you to, why bother? As long as leak stopped, floor dried and no mold condition certified.)
You don't need to give them money to move. Just don't renew the lease. (there should be another item about early termination from both side)
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