The warranty is counting for each device you own from the moment you bought that device. You can also check the warranty status for each device here: https://selfsolve.apple.com/agreementWarrantyDynamic.do
Before you contact Apple to get your phone looked at, try to set it up as new device, without using the latest backup afterwards. This way you can rule out a software issue.
If the phone is still not working as expected, contact Apple Support or visit an Authorized Apple Service Provider or Apple Store next to your place to get it serviced.
If you used your backup to get your data and settings back, there could have been corrupt files inside, causing your problem. To rule that out, set it up again as new and see if it running without problems.
Setting the device up as new is the first thing they would do at the Apple Store to make sure the iOS software is not responsible for the problems your device has.
iTunes and App Store content
iTunes Store and App Store content on your iOS device may not automatically transfer to your computer. You can use the Transfer Purchases option in iTunes to ensure that all iTunes Store and App Store content is transferred to your iTunes library. Learn about transferring purchased content.
For App Store apps that use File Sharing, transfer the files from within your apps to your computer using iTunes.
To keep pictures or movies you have sent from the Messages app or received in the Messages app, save them to the Camera Roll (or Saved Images library on iPad and iPod touch). Then import them to your computer.
You can import photos or video taken or saved on the iOS device. Information on importing photos and videos is available in the iPhone User Guide, iPad User Guide, or iPod touch User Guide. You can find additional information on importing your photos from your iOS device in this article.
Note: You can import only photos or videos that you took or saved on the iOS device. These photos will be in your device’s Camera Roll (or Saved Photos on iPad and iPod touch.) You can also share photos with others through emails and messages.