Always backup your data. During normal use, and if you need to do anything extraordinary.
Apple have that Time Machine function for backing up, which is nice.
Personally I prefer to make a backup on an external hard drive (usually connected by USB port) by just dragging and dropping the entire data/documents/photos/etc content of my drive onto the external drive. You can also choose to copy any settings files or other material that may be useful if you should need to reinstall anything at some point.
You should do this routinely anyway to be safe.
There are cloud-based services that offer backup.
Generally though, changing a battery should not affect your files/data or installed software or system. These reside on the hard drive and it does not need power to retain that material.
Don't know about Apple store location inventory, just phone them I guess.