IPhone – If we run out of battery and then charge iPhone, will it turn back on

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.

In a word: Yes.

To the best of my knowledge, the mere act of plugging-in the dead iPhone to a recognized charger will cause the phone to initiate the boot sequence once the battery has charged sufficiently to allow for running the phone and charging simultaneously.

You can check this behavior by turning your phone off and then plugging it into a known-good charging system. You should see the apple appear shortly and the phone will proceed to boot-up.