COE means close of escrow. That is the term used for completion of the contract, money and keys transferred to the appropriate parties. From the information you have provided in the post, the "10 days" could mean they want a short COE (normal is about 30 days). It could also mean when possession of the home will occur. In other words they want to stay in the house 10 days after COE . This could trigger a "rent back" clause from the buyer. In other words, "Yes, seller, you may stay in the property 10 days after COE but you will owe me a daily rent of x."
If the seller wants to close 10 days after an offer is accepted, that may be a challenge for most buyers. Check with your lender to see how quickly they could pull the paperwork together. In the past, when the market was a seller's market, many lenders could close in 14 days. Now with all the changes in underwriting as a result of the sub-prime issues, I do not believe that is the standard.
Also, in California, the standard time for completing inspections is 7-14 days. If the seller is pushing the time line, you need to be sure you can complete the inspections necessary to feel comfortable with releasing the physical contingency as well.