Cannot reinstall Fairphone OS or Fairphone Open OS - only Lineage

That would indicate a hardware problem I think.
The only thing I could think of otherwise would be tweaking CPU settings to perhaps work around a performance or heat problem … e.g. with Kernel Adiutor or with LineageOS performance profiles.

If you happen to have fairphoneangels in your vicinity, they could perhaps help by swapping modules to find out whether a module is at fault, even if the expensive core module seems most likely to me.

Else … install one of Fairphone’s OSes and contactsupport

… and quickly for a possible warranty case, if it was the end of 2016.

Can you give more details on that?
WiFi works fine for me.

Edit: Wait … does LineageOS not do the reboots?