Problem with iPhone Battery

Having problems with iPhone battery. First of all, it must be re-charged almost constantly even though I keep the LCD screen on medium brightness. Often when re-charging, the phone gets extremely hot but the biggest problem is that the phone loses energy quickly on cold days. I keep the phone in my pocket to keep warm but it still goes dead on cold days after removing from my pocket. This is unacceptable as I don't receive important calls nor can I make calls on cold days. Is this normal with all lithium-ion batteries and with all phones or just with the iPhone battery? It is very frustrating.

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I want to know, whether can I sometimes use not original charger for iPhone? Or will it harm the battery?