Charging/USB doesn't work except when turned off, phone reacts to USB-cable but crashes seconds after charging begins

[Volker]

Well, 30% is not much. I’m not sure it’s enough for a backup…

@fnadde42

Can you recommend a tool (phone is not rooted)?

You might try MyPhoneExplorer to copy some personal data.
Or you could do a backup using fastboot

FP2 How to use TWRP without flashing it (e.g. for Backup)

I already posted it here, but I’m pretty sure many people aren’t aware of this I found a way to boot TWRP without flashing it, so you only boot it from your computer, do what you want and as soon as you shutdown your phone, it’s just as it was before. TWRP is a great tool for many things (install updates, perform a factory reset, create or restore a (full) backup, copy files between PC and FP2, …) It’s pretty simple to do this, the hardest thing may be getting fastboot …

You also could have a look at the support pages:
https://support.fairphone.com/hc/en-us/articles/210512123

But in any case you should open a ticket with support as your phone is still under warranty (if bought new in summer).