I have an iPhone 5 on iOS 7.1 and have had none of those problems.
Something may have gone amiss in the upgrade to iOS 7.1 on your iPhone.
To correct this follow these steps:
1. Backup your iPhone using iTunes or iCloud.
2. Use iTunes to completely restore your iPhone as Factory new.
3. Monitor behavior to see if the iPhone is stable. If not, you may have a hardware problem - if so, contact Apple support.
4. Restore your iPhone from your backup above.
5. Check stability to confirm that it remains normal.