Battery

Apple does not consider that an iPhone's battery requires replacement unless it cannot maintain, at least, an 80% charge after 500 full recharging cycles. However, it may be possible that this iPhone's battery is defective. As such, I suggest that you make a Genius Bar appointment at your local Apple Store or Apple Authorized Service Provider to have them take a look and provide you with your options.

No, but it would depend on which iPhone model you have. If you purchased it new, directly from Apple. If you had the battery replaced.