I can't add a new contact as keyboard doesn't appear in the Contacts. I have deleted email account from Settings. Also the Contacts service is on in iCloud.
I have an iPhone 4 on iOS 7.
what do you mean you deleted email account from settings? Did you un-sync an email account which is needed for the contacts? Did you just reset your phone when you try adding contacts?
I have iphone4s and I never had a problem like this when adding contacts(I'm pretty sure iphone4 and 4s aren't so much different from each other).
Try to make sure you have synced an email account first in your settings>Mail, Contacts, Calendars. About icloud7, it's turned on in my phone but I haven't had problems adding more contact. iCloud just syncs your things in your phone and it would make any issues upon adding one.
Reboot your phone. Mine just experienced the same problem seconds ago. Turn it off and restart, then it should be fine.