Product registration and warranty??

You can find the serial number: Apple Menu / About This Mac / More Info / Hardware Overview.

Call Apple Service, or go to an Apple Store, and give them the number. Warranty info is stored by machine serial number.

I have no idea how you would determine if the computer is "hot".

Registering the computer in no way affects the warranty. Apple likes to collect marketing information on people but the warranty is in effect, by law, regardless of any "registration".

Now, to address you question specifically. Apple reserves the right to require a proof of purchase in the event of warranty work. This goes for AppleCare also. When you register AppleCare (the extended warranty), they simply ask you for the date on your bill of sale, or receipt, with the disclaimer I mentioned above. I recently had warranty done at an Apple Store and they did not ask for the invoice, although they could have and refused to cover the work if I could not prove the date the computer was purchased.

So basically, you're out of luck if you need warranty work AND they ask for proof of purchase. If you purchase AppleCare, you have no idea what the "original" date or place of purchase was without an invoice and if you needed to have work done AND they decide to ask you for proof of purchase you're, again, out of luck.

Now, I can't say under what circumstances they're told to ask for proof of purchase. I imagine if the machine is an older model and they may feel the warranty questionable, I really don't know.

I recently tried to look up the machines and iPods I registered in the past and could not find it. It does not appear they offer that information and I would guess because machines and iPods are sold all the time. The only thing you can look up is your AppleCare plan tied to your "user ID" or "account name". If a covered (under Apple Care) machine is sold, you have to get Apple to transfer the extended warranty to the new purchaser's name and address. Note: you do not have to "register" your computer when you first purchase it. After you set up your user name/account for the machine, you can quit the registration process and transmit no information to Apple about that machine; it is purely for marketing purposes and somewhere in the fine print you'll find "registration does not affect your warranty rights".