That is strange. It is important to keep you address book up to date. Let’s look into this further. Do you have your contacts synching with an email account from your work or school? How did the contacts get imported into your device? Can you add a new contact and then delete that one?
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Have you tried resetting your phone (hold power button and home button simultaneously until the apple appears on the screen)? Sometimes that helps when random glitches occur.
The only thing linked to my contacts is Facebook. If I add a new contact and try to delete it, there same problem occurs. Most of my contacts were added manually though.
Check to make sure restrictions isn't on for contacts. If it's on and "do not allow changes" then your edit button wouldn't exist because you're not being allowed to delete. Not sure if this is your issue but it doesn't hurt to make sure.
Only other thing I can think of if calling Apple isn't an option (not sure if you are still under warranty) is to try restoring your device as new and seeing if the issue still exists. Make sure you backup your device either via iCloud or iTunes. Before putting you backup back on your device try using it without the backup and see what happens when you add then delete a contact. If it erases then add your backup back on and test it again.