Which of these three values does the bank actually use?

The lender cares about the appraisal and nothing else on your list. The assessor's number is for property tax, it's produced on a mass schedule for thousands of parcels at once, and in many counties it's a fraction of market value on purpose because the tax rate is set against that base. It says nothing about what the house is worth. The Zillow number is an automated valuation model, a statistical estimate from public and listing data, which is a real tool that lenders do use in some situations but not usually for a purchase mortgage.

The appraisal is a licensed person walking through the house and writing a supported opinion. That's the one the loan gets sized against.

On the low appraisal question. The lender lends a percentage of the lower of price or appraised value. If you're putting 10 percent down on 279 and it appraises at 265, the bank sizes to 265, so the gap comes out of your pocket unless the seller drops the price or you split it. The bank doesn't refuse the loan, it just refuses to lend against the extra 14k.

Whether your contingency lets you walk with the deposit depends entirely on how that specific paragraph is written and on your state's contract forms, so have your agent read you the operative sentence and, if there's real money at stake, an attorney. Some versions give you an outright termination right, some only give you a window to renegotiate before the right expires.

The 550 is generally non-refundable once the appraiser does the work, which is the part people are surprised by. You're paying for the report, not for a good outcome.

The thing that finally made this click for me: assessed value, an automated estimate and an appraisal are three different jobs done by three different parties for three different purposes. They have no obligation to agree and they usually don't.

The one I'd stop looking at entirely is the Zillow number. It's an estimate built off recorded sales and listing data, and in a dozen or so states sale prices aren't public at all, which means those models are working with much thinner input than people assume.

@verity that answers something I'd been wondering too, thanks. @juniper the 14 day thing is the part I'd move on first. Ask your agent today what date the appraisal is actually scheduled for, because if the appraiser isn't booked until day 11 and the report takes five days to come back, your contingency expires before you ever see a number. That happened to someone in my office and it wasn't discovered until it was too late to extend.