It's a scam. Or they're really dumb.
A bona-fide company never needs your credit card details, certainly not your 3-digit-on-back-of-card #, to issue a refund. On an older charge, they might have to work with their merchant provider. But they should be able to do it within the credit card handling system, and in fact are required to.
Asking for details via email doesn't pass the "sniff test" either. To get a credit card merchant account, a company needs to go through a security assessment process called PCI-DSS. Security gets drummed into you pretty good. Of course they could be using one of the dumbed-down services like Square, but those services make refunds ridiculously easy.
How did you come to be corresponding on this email address? Did they initially contact you? Did you find it on a third party website? Some of those are fraudulent and many others, like Yelp, it's very easy to insert false contact information for a business. Consumer forums, even moreso. You might take another swing at finding a proper contact for the company.
Stop asking for a cheque. That also circumvents the credit card system. And obviously a scammer won't send a check... at least not one you'd want!
If all else fails: call your bank and tell them you want to do a chargeback on that transaction. This is where the bank intervenes to reverse the charge. It's rather straightforward (especially if the merchant has agreed in principle to a refund) but requires some paperwork or e-paperwork.
Don't chargeback lightly. Don't use it casually or out of laziness or unwillingness to speak with the merchant, e.g. to cancel an order. The bank charges the merchant a $20 or larger investigation fee, separate from the refund. Each chargeback is also a "strike"; too many "strikes" and the merchant is barred from taking credit cards. It's serious business.
As a merchant, I would never send a cheque to an angry customer. Because if I did, they'd cash the cheque and still do a chargeback, so then I'd be out the money twice, plus the investigation fee to boot.