I would question two things about your framing of the issue (and will sidestep the whole repair discussion/issue entirely):
1) Apple's warranty does cover batteries that don't retain a percentage of their useful life based on aging at one, two and three years (two for most iOS warranty / coverage, but there are exceptions in some cases). You should open a ticket with them if you have coverage and see if they will repair it for you.
2) I use an iPhone regularly outdoors when the temperature reaches -30 C and even then, the decrease in capacity is temporary. While the device is colder than 0C - you get less juice out of the cell, but once you warm it (in a pocket, next to your skin to warm it temporarily - or when you are inside a heated space for long enough to warm the battery), the devices recover full capacity.
(You will want to watch to control condensation to prevent liquid damage / especially if you bring a very cold device into a warm humid space, that warm air will carry liquid inside the product. Similarly, a warm device when rapidly cooled could have liquid condense inside it and then freeze.) Devices left in a car overnight such as iPad and Mac also see no measurable decrease in battery life once I get them inside and warm - whether I charge them or not.
I'll leave the DIY repair / pay for a repair discussion for another thread where you get specific on what model, but from what I see, the environmental behavior is that cold batteries still have all the charge they took, but they don't produce as much voltage and current while they are cold.
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.
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.
Best Answer
If you plug an iPhone into a charger it should power on. If it's on but frozen, you could connect to iTunes and reset/ wipe it, but that might lose some data depending on your backup situation and what is on the phone.
If it doesn't, then there is a hardware problem and you'll need to get that fixed or live with a faulty device.