How to bypass iCloud login for iPhone 5 running iOS 7 when email and password was forgotten? I restored my iPhone 5 with iOS 7 without knowing that iCloud account will be asked. I already forgot what email and password I set on that phone.

Now, after the phone was restored and unlocked but no application can be used. I don't know what to do. When I connect it to iTunes, it keeps on asking for the iCloud email and password.
Since you can not bypass the activation lock itself, you need to contact Apple and get control of that account. If you are the original purchaser of that hardware, you might also be able to get them to reset the lock with proof of purchase and identity. 

The website https://appleid.apple.com has instructions on resetting your iCloud password if you do not remember it. 

You can retrieve the email by erasing the iPhone from iTunes when the device is in normal recovery mode and you will be prompted to enter the password for the email account. Also, if the device is not locked, go to the settings app and look in iCloud for the AppleID that has locked that device for firer activations.