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Generally you set up the closing with your LO or Broker. If the date they set up is not convenient for you, just let them know. Then let them know what day would works best for you. Usually is not a problem to change closing date with sufficient time notice. But as always YMMV
When I sold my house, the buyer picked the closing date and I said ok thats cool. I gave them 60 days.
Yes, here too. The buyer selects the date in the contract offer. We keep in mind the amout of approval time the lender needs in order to get the financing package completed. If the seller doesn't like the date, the time to change it is at the offer stage.