The easiest way to transfer the content from one iPhone to another is to backup the old iPhone to iTunes on your computer or to iCloud and set up the new iPhone from that backup.
Yup, what scifan said. It's by far the easiest and best way. Your 7+ afterwards will be just like your 5C when it turns on after restoring from 5C back up, just bigger, badder, faster, etc.
Not yet. My battery wigged out once, and that's what brought me to this forum, but it was a one time thing and it never did it again. I never took it in for repairs or anything. Just happened the one time. Phone is great and I love it. My 6+ was always kind of on and off glitchy, but not my 7+. Never gives me problems.
Remember to do an encrypted backup - write down the password, you'll need it - so your passwords, login info, etc are backed up and restored to the new iPhone