Really need your help about the 1 year warranty!

Well 2 weeks ago I brought a used iPhone 5 and I thought it was great for the first day and after that I only just realised how awful the battery is and when I say awful I truly mean AWFUL! You can literally watch it go down.. I have done everything in the setting so don't say change all of that stuff because its the battery! The charging is rapid but I think that iPhone 5's are fast in general. From nothing to full it probably takes 50-60 minutes. But it drains so quick it's rather scary.. But whàt I don't get either is that sometimes it's on 12 or 21% and it dies and says it needs charging then about 20 minutes later you can turn it on and it will be on same battery life? The phone was brought for a Christmas present in December he said ad should still have warranty but will they accept that? Another big thing is today I dropped it from 3 foot onto rather soft carpet and the screen smashed! I was gob smacked I couldn't believe my luck! I just hope they don't think I'm trying to get that raplaced for the screen..

Will they accept it and do they accept used phones (if they have a way of finding out)?

Thanks so much for your help!