Vaguely sounds like a defect motherboard to me. Getting the data off it would be quite difficult then – probably not without a professional recovery service, perhaps even tricky for them.
You can look for a working (used) motherboard here in the unofficial marketplace and/or consult a Fairphone Angel (might have a spare motherboard)
Should you decide to give up on the phone, please donate or sell parts needed by others (especially the display unit). Fairphone Angels also take them.