Sony Xperia Z3 Compact User Manual

Take it to a Sony service center for replacement. You don't want your phone to be repaired by service center other than Sony's as you may get problems with it's rugged features. You can contact Sony first to know how it will cost and also ask for the nearest service center in your area.

I need to replace my Sony Xperia Z3 Compact battery. How much is it and where do I buy it?

Since the battery is embedded, it is best to take the phone to a Sony service center to get original parts and to avoid removal of warranty.

Check if your phone has a Signature feature in the text messaging app. If there's none, use a 3rd party text messaging app, most if not all, have this feature.

While taking a video, tap the camera icon to take photos. Please let me know if that works.

Hmm.. There's probably an option there. When in homepage of the app, try pressing menu.