how old is my ipad

Your device serial number cannot be used to provide a definitive date of manufacture. The best that can be offered is to verify your estimated warranty coverage - the date of which will correspond with the date of initial activation.

https://checkcoverage.apple.com

Apple won't actually replace the battery in your iPad, but for the fee of a new battery, Apple will give you a replacement (refurbished - actually indistinguishable from brand new) iPad with a new battery in it.

The fee is listed in this link --> iPad Repair - Official Apple Support

As you can see, near the bottom of the link it will cost $99.

That said, when iPadOS 16 is released in the fall, it will not be offered to iPad Air 2 devices. Your device is now at the end of its supported life and will be frozen at iPadOS 15.x. Only you can decide if putting $99 into an older device which loses support soon, is a worthwhile expense. It might make more sense to put that money towards a newer, more modern iPad. But that is your choice to make.

iPad Air 2 was released publicly October 22, 2014. It is your responsibility to save your purchase receipt. Not Apple's. Regardless of whether you purchased your iPad in 2014 or 2016, it's an old device at this time and if the battery is the original battery, it is doing what batteries do and that is lose their ability to hold a full charge as they did when new.

I honestly don't see what your point is? Batteries are consumable parts and wear out. And devices get old and technology passes them by.