IPhone 6: battery replacement and display problem

Apple does offer a one year warranty, so if you are the first buyer, you should be able to register and get them to replace the battery for free. I don't know exactly when they discontinued selling that model, but start with trying to register that device to you at support.apple.com using the serial number from the box. You might need to call into support or fax them the receipt since it's so far past when they were expecting to enroll new owners, but once you know if Apple can/will service it, you can decide to repair it through them or on your own.

It's likely the lithium battery has failed permanently due to not being stored for long term with 50% charge. I typically get out my old devices once a year and charge then to 50% and then power it off. If the voltage drops too low then you will need to replace the battery as it will never charge or even hold but a fraction of the intended energy.

The customer care has given you good advice and it's great that they offered you the money back. I would say if you are happy with the price, look at repairing the battery yourself. That design was very easy to self-service - especially if you realize it's unique and will take your time with the service.

• https://www.ifixit.com/Guide/iPhone+3GS+Battery+Replacement/1545

The parts should be readily available and the tools are inexpensive. Older phones (original) and newer phones are much harder to replace the battery due to tools or design of the connectors and internals. I'd say keep the phone unless you really can't afford the replacement cost of the battery.

It appears to be the case that Apple will replace the top case assembly if you want to replace your battery.

As first reported by Treehugger, which received a copy of an Apple internal service document for the "Mid-2012" 15-Inch Retina Display MacBook Pro, Apple informs its own service personnel that "batteries must be replaced with the top case assembly" and that "the battery alone is not a replaceable part." EveryMac.com also has been provided with copies of the internal service documents for subsequent 15-Inch Retina Display MacBook Pro models and although there are internal differences, the battery design effectively is the same.

Additionally, Apple warns its own technicians that "batteries should not be separated from the top case assembly for any reason," as attempting to do so could puncture the battery and lead to fire or injury. Do not attempt to pry the battery apart from the top case.

Source: EveryMac.com

According to iFixit, it will cost local repair shops around $500 to source the top case assembly to replace the battery.

IFixit.com also has instructions on how to remove this battery should you want to replace it. Batteries with a minimum of 12 month warranties can be had for $90USD makes this a worthwhile consideration. Keep in mind that this is a difficult repair and if you are unsure of your ability to tackle something like this, best hand it over to a pro.