Is there an app that can back up (and restore) all files installed on an android smartphone, including apps with their settings and data, contacts, messages, etc. ... preferably with only one click?
There's no one click solution. For a total backup of *everything* on the phone you can do what's called a nandroid backup, but that requires rooting the phone.
Going to: 'Settings > Backup and reset' and enabling 'Backup my data' and 'Automatic restore' is a good idea but by no means is it a complete backup of everything on the device. That will transfer your Google stuff like contacts saved (if saved as Google contacts), calendar, all compatible apps and some system settings.
Documents, photos, music and videos can be transferred with a PC and USB cable, cloud storage or WiFi with Airdroid or similar apps.
For the rest, such as SMS/MMS, etc MyBackup Pro works well and doesn't require a PC. Helium is better in that it may save more settings and other data than MyBackup but still maybe not *everything* and it does require a PC.