If it's not even to the underwriter yet, then no closing date can be set. You may have been given a TENTATIVE closing date, but that really means nothing. The underwriter can ask for more information and can ultimately deny the loan. When you get a firm commitment, that's when the attorneys can schedule a closing (assuming all of the title work is done).
When you get the commitment letter, that is the time to get your homeowner's insurance policy. DO NOT, after getting the commitment, think that you are all clear. Do NOT go out and finance ANYTHING. Not furniture, not anything. People have gone out and bought furniture or appliances or even cars and financed them, and when the lender did a last-minute credit check and saw these things, it blew up their mortgage, they were denied, they lost the house and they lost their earnest money. And in some cases, the seller sued them for expenses of the sale that had already been paid, and the seller won.