Who pays the utility bills and property taxes for the days right around closing?

This is handled through the statement of adjustments, the calculation lawyers prepare to apportion shared costs as of the closing date. Utility charges, and property taxes if real property is part of the deal, are split so the seller is responsible for the portion of the billing period up to closing and the buyer is responsible from closing forward, regardless of when the actual bill happens to arrive.

Because utility and tax bills often cover a period that straddles the closing date rather than starting exactly on it, the adjustment is usually based on an estimate at closing (using the prior bill and a daily rate) with a true-up once the actual bill comes in, if the agreement provides for that. This is a routine, mechanical part of closing rather than something either side needs to negotiate case by case, but it's worth reviewing the actual statement of adjustments your lawyer prepares to confirm the numbers and dates used match your understanding of when closing and possession actually occurred.