Sorry in advance - this is going to be long:
I've been keeping an eye on this thread hoping someone would have come up with a solution by now. My husband and I both upgraded to the iPhone 7 the first day it came out from the iPhone6. He got the 128GB black and I got the 128GB rose gold. We both restored from our backups. His phone has been fine from day one. Mine has been a nightmare. Not less than 24 hours after having the phone, the keyboard kept freezing up, and then the entire phone was frozen so badly that I couldn't even get it to reset. I brought it back to the store and they returned it under their return policy and gave me a brand new phone (iPhone 7 #2). I restored this one from my backup as well. Within about 3-4 days of having this phone, I began having the issue of the keyboard freezing in messages, emails, notes, etc. I hopped onto Apple chat and the Apple tech had me wipe the phone and set it up as new (via a computer with iTunes), assuming that something in my previous backup was corrupt and causing the issue. I pulled in my contacts from icloud/gmail, and downloaded my previously purchased apps. The issue continued to persist. I then called Apple support, and the woman I spoke with suggested I go into an Apple store and speak to a specialist. This was on 9/21 and the Apple store specialists had not heard of anyone having this issue yet read the case history and replaced the phone under warranty. I again set up the phone as new out of the box. The issue still persisted, and so I made an appointment to go back to Apple on 10/3. I spoke with a different specialist, and he remembered me from the previous appointment and read my case history again. He spoke with his manager, and said that they don't believe it's a hardware issue since it's persisted across multiple devices. He said to wait until IOS 10.1 was released and that if that update didn't solve the problem to make another appointment and come back. Needless to say, 10.1 did not solve the issue, and I was back at Apple on 11/4 and got iPhone 7 #4 under warranty, yet this one froze up so quickly that I was back at Apple again on 11/5 and they replaced the phone yet again and strongly suggested that the issue be escalated to a senior advisor at Apple. So I'm on my 5th iPhone 7 and am in direct contact with a senior advisor at Apple. It's now been escalated to Apple Engineering. I've wiped my phone again and they're trying to isolate the issue. I was out of icloud for a couple of days and the issue didn't persist. I've now signed back into icloud but have no apps on my phone, and I just signed back into iTunes. I'll update as this progresses, and hopefully with a solution.