Ex-Apple employee here (4 years).

Yes, you can absolutely return it to any Apple Store (without the receipt) within 30 days of the purchase (14 days is the standard policy, but this is a guideline rather than a rule, though sometimes the return date is longer than 30 if it was purchased around a holiday. Apple is pretty flexible). They can look it up via serial number or email address (which older people generally don't use/have/remember).

As long as the product was in the original box/packaging undamaged and with everything it came with, you should be fine.

As for the return currency, it'll go back to your grandmothers original card, or if she paid cash, you could get the cash, though that high amount will likely go to a gift card so they can track the return for safety of all parties. This varies in the employee doing the return and the manager working that day, however.

If they picked it up at Best Buy, the above should be possible, however I can't confirm as I never worked there. Many have apple designated representatives that should be able to process the no-receipt return the same way though.

Hope this helps!

Edit: It should be noted that this is true for nearly all serialized products as apple has a database to keep track of all serialized purchases. Their system even allows them to verify if the product was purchased outside of apple retail (the system shows if it was purchased at Target, Best Buy, Mac Mall, etc.). With the above said, the same is not true for products without serial numbers, as there would be no way to prove if it was purchased at an Apple Store and thus no way to track it.

TL;DR - I am known for unnecessary long-winded answers.

Edit: it should also be noted that the 14-day policy is a guideline, and that 30 days is typically the cutoff point. Keeping that in mind, many employees can make exceptions beyond 30 days depending on the circumstances. I personally returned products that were well-over 30 days if I felt it was necessary, though not all employees have this ability as its a system override/trust granted to them via the managers. 14-day is a guideline, not a rule.

My friend won an iPad mini at work. He took the unopened box to the Apple store and "upgraded" it for an iPad mini Retina by simply paying the difference.

Is the one you received the same model as your existing one as the new Apple TV 4 was recently released and has quite a few new features. Do you have more than one TV? Many of us have and use more than one Apple TV in a household so you could consider doing that. If you really do want to return or exchange it then check this article or call your local Apple Store and ask about their gift return policy. http://www.imore.com/what-you-need-know-about-apples-holiday-return-policy