Technically, your receipt is all you need to prove you bought the product, when and where. So legally, you do not have to register your product to activate the warranty and obtain warranty support. But registering can make getting support faster since the purchase will be on record. That can be a lifesaver if you lose your receipt.
Registration also gets you on their list for notifications in the case of product recalls, or of new driver and firmware updates.
So in your case, with your mouse and keyboard, those are pretty low-tech, reliable products Windows already knows how to use. Your printer, depending on its features, may be worth registering.
HOWEVER! Submitting your warranty information is also a good way to get...