Can I create/edit contact groups in iOS?
No, there is no way to create or edit contact groups on iOS. I share your frustration! The best workaround is to use iCloud.com (or your primary Mac, if you have it with you).

Also, there is a way to add contacts to a specific group on iOS, though it's not obvious:


In Contacts, tap the Groups button in the top left
Tap "Hide All Contacts"
Tap the group that you want to add the contact to (making it the only visible group)
Tap Done in the top right
Tap the Add button (+) in the top right


Now when you create and save the contact, it'll save it in that specific group. You can then go back to the Groups list and reenable all your groups.