When you buy a new AP ahead of time, you can't just walk into a park until you have had your AP activated. (You need to activate at a ticket booth or at a Guest Relations at the parks or Disney Springs - resorts don't do AP activation.) That will cause the clock to start running on your AP. If you bought it on June 15 but you don't activate it at WDW until September 23, your pass will be valid from September 23 through the next September 23. When you activate, they'll give you an AP card. You need to keep this with you, as you will need to present it to redeem AP discounts. Magic band scans don't count as proof you have an AP for this purpose.
Think of the magic band as a key to unlock your MDX account. There's nothing stored on the band. All it does is access the info from your account. If the AP is linked to your MDX account, any magic band registered to you on that account should be able to read it. That's why you can reuse old bands.
As for the PP's aunt and uncle, before using any of these tickets, you need to assign your aunt and uncle's tickets to them so the system does not think the tickets are yours. If they don't have an account, I believe you can add them to yours as family or friends. They will need magic bands or ticket cards assigned to them to enter the parks via the tapstyles.