Yes. They can access your purchase scanning your membership card
Costco keeps an electronic record of your purchases
I think they will honor purchases within the past three months without receipts, beyond that it's their discretion. Jewelry however may have a different set of rules, better check with them first.
They can go back into your history to see when you bought things from your member card.
Costco has a generous return policy and based on your membership you should be okay.
I have returned general merchandise without a receipt at Costco. All purchases for the recent months are identified when they run your membership card number. There is no harm in trying and talking to them. I am not sure about the return policy for jewellery item such as the ring.
The best way to know is to go to customer service and ask. Or call.. Bring any documentation you can, like you credit card charge if you used a card.
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