OK, this is going to be a loooong response lol.
First, I believe there is a one bag per passport policy at Chanel. (Note that WOCs are considered slg's so not part of the one per passport limit so you could buy one flap and one WOC per passport without any problems.) Also, I doubt that Chanel has a way of checking if you bought a flap at Chanel Bond Street and then bought another flap at Harrods or at Selfridges, for example, but just be aware that you
might have to split your purchases between locations around the city. If you are buying seasonal items, I think they are more loose with the restrictions.
My last check of prices indicated that a small classic flap was £3,810. The VAT refund should be roughly 13.4%. (This was my exact % refund from London when I bought my small classic flap there a couple years ago. Yes, i have a spreadsheet.

). So, your VAT refund should be roughly £510 with a net cost of £3,300, converted to $4,290, as compared to $5,000 in the U.S. plus sales tax in whatever state you live in. So, a reasonably significant savings.
U.S. customs duty/tax on your purchases when you re-enter the U.S. will depend on how much you bought in total and how large your traveling party is. Each member of your traveling party gets $800 tax free, next $1000 is at 3%, amounts over $1,800 are taxed at 6%. So, if you are two people traveling: $1,600 is tax free, next $2,000 only costs $60, then the balance over $3,600 is taxed at 6%. So, you might not pay that much unless you are coming back with tens of thousands of dollars worth of small classic flaps.
Your second question: you claim the refund at your last stop in the EU before returning to the U.S. If you place your items in your luggage and check your luggage all the way through to the U.S. in Athens, then you would get the Customs stamp and complete the process in Athens. If you keep the items with you and carry them on the plane and have only a very short layover at LHR, then you would still want to complete the process in Athens. However, if you carry the items on the plane with you and have a long layover at LHR, then you could wait to do it at LHR. (The items have to be in their original packaging and available to be inspected and then the paperwork will be stamped by Customs so if the items are checked in your luggage at your starting ponit, then you have no choice but to complete the process in Athens.) However, please, take my advice and keep your items with you and carry them on the plane. It is worth the hassle. You don't need them to disappear or be damaged in transit. If you choose to do the refund on your way through LHR, make sure you have
lots and lots of time. The lines can be long and you want to have plenty of time to complete the process. Like hours of extra time, just in case.
Good luck finding a small classic flap, they are tough to track down and you might have to call everywhere in London, but it's worth it. I love mine!