looking for any kind of damage (not only ransomware)
Detecting if the entire file is encrypted is trivial (there's even a MSWindows version). However something which encrypts the entire filesystem would be hard to detect automatically (but the impact very obvious).
Something which encrypts or damages partial content inside a file would be very difficult to differentiate from a normal user edit (lots of file formats are actually containers with multiple individual files inside - e.g. Adobe PDF, Microsoft Office). Analyzing the frequency distribution of characters (or tokens if you have a parser) can give more heuristic detection if the file is not compressed.
There is a lot of activity aimed at detecting ransomware by its pattern of file access (frequency of file renaming, file changes, directory crawling access).
A fairly simple solution is to monitor a small collection of files in the root and at the bottom of the directory tree (i.e. the places where malware will strike first) on a dedicated share as canaries. If these files are updated (without action by a user) then you know there's probably malware at work. Any host based IDS should be capable of this.
Ordinary backups are not really secure, because the ransomware could just as easily target your backups
If your "Ordinary backups" are available online then yes - but that's why we use tapes, optical drives and virtual tapes.
Is there any existing product
"Questions seeking product recommendations are off-topic as they become obsolete quickly"
(a) automatically make copies of designated files folders
Yes, there are lots of file copying and backup programs.
audit the live files against the archive