If the mouse becomes out of order, would the number starting with CC on the back of the mouse indicate to the Apple employees whether that mouse came with an iMac that has AppleCare+ or not?
Generally speaking, Apple accessories such as keyboards, mice, trackpads, power adapters, charging cables, and cases do not have serial numbers like standalone Apple products do (Macs, iPhones, iPads, AirPods, Apple TVs, Apple Watches, etc).
As a general rule of thumb, if your Apple product has a serial number that can be looked up on https://checkcoverage.apple.com/, it has a warranty period (Apple Limited Warranty and/or AppleCare+) that can be verified by Apple. For products that do not have serial numbers that can be looked up, Apple will generally take your word for it if you say that you purchased the accessory less than 1 year ago (Apple Limited Warranty) or that the accessory came with a product that is covered under AppleCare+ (in this case you will provide the serial number of the product covered under AppleCare+).
As long as your story is reasonable (don't try this with an accessory that was discontinued a while ago) and you don't mention accidental damage, you should be able to get your accessory repaired (most likely replaced) for free. In my experience, you'll likely have much better luck with this over the phone via mail in repair/replacement rather than taking your defective Apple accessory to your local Apple Store.
Generally speaking, Apple accessories such as keyboards, mice, trackpads, power adapters, charging cables, and cases do not have serial numbers like standalone Apple products do (Macs, iPhones, iPads, AirPods, Apple TVs, Apple Watches, etc).
I can't speak to the recall program, but if you have a defective power adapter or charging cable, you can contact Apple Support and they will send you a replacement. You authorize a hold on your credit card equal to the value of the accessory. When you receive the replacement accessory, you use the provided packaging and prepaid return label to return your defective accessory. Apple releases the payment hold when they receive the defective accessory.