Recovering from run time errors on stm32l0
Yes there is normally a bit that will show you power-on/reset cause (have you checked the datasheet?).

One way to catch the errors might be to configure watchdog as interrupt instead of reset (not sure about STM32F0, but I have used this on similar chips).

As for saving and restoring RAM to/from EEPROM - it might be easier to just dump RAM contents in the reset handler (before CRT clears .bss and copies .data sections).