Hello melissafromvirgie,
We see that you have some questions about Live Photos. We want to help you locate the answers you need. To start, are you following these steps when you Take a Live Photo in FaceTime on iPhone? If so, and you still have the issue, Restart your iPhone. This simple step can help to refresh the apps and software on the iPhone and help to solve numerous problems you may encounter.
Another thing to check would be if the iPhone is up-to-date. You can go to Settings > General > Software Update, and check to see if there's an update available. If so, make a backup of the iPhone and update the software. Here are details on How to back up your iPhone, iPad, and iPod touch and Update your iPhone, iPad, or iPod touch.
Cheers!