Argh.. I thought it was just me. I restored my phone and reinstalled iOS7 thinking I got a bad copy.
My iTunes Radio is working but I'm getting the exact "Unable to connect to server" error with iCloud. It started when it asked me to Agree on a new iCloud Terms of Services. And it won't Accept because it can't connect to server.
My friend on the other hand cannot connect to both iTunes Radio and iCloud.
Another friend is having intermittent problem with iMessage.
So I guess it's Apple servers and we just have to wait until it settles down.
At least we know it's not our devices and/or apple accounts that are faulty. This is somehow a relief.