What type of damage can power surges do?
[I]s it possible for power surges to inflict partial damage on electrical devices?


What you refer to as a power surge will most likely be a voltage spike, i.e. short but significant increase in the line voltage. This increased voltage will in the most general case lead to increased electrical currents in your devices, and this may kill them.

Simplest example: a light bulb that is specified for 230V will generate more light at higher voltages (because higher voltage -> higher current -> higher temperature of the filament -> more light), but due to the higher temperature, the filament will break earlier (as early as "instantly" if enough voltage is applied). Other simple examples might be motors which might run faster at higher voltages but will also suffer from the increased currents.

Any devices involving semiconductor circuits (so, almost all nowadays, I guess) are also affected in the same way. Too much current -> too much heat -> broken device. It is quite possible that only a part of a device is affected and the consequences might vary from fancy noises to complete failures.

So yes, it is possible


  and if yes, is it likely?


That totally depends on the nature of the voltage spike and the device.


  I'd like to know if those events are correlated together and to the
  electrical variability of the apartment


There's no way of telling this for sure. It might be the case that the dying fridge was causing yet another voltage spike that further damaged the microwave oven, or the events might be completely unrelated.