Electrical – Monitor blacks out when fridge turns on – bad power supply? Surge protection

Breakers and fuses are sized to limit the fault current that can flow. Each is sized to limit the current to that which the cable connected to it can safely carry. For example a 32A ring main circuit will protect a 2x 2.5mm2 cable ring circuit. Another example will be the 6A lighting circuits. These could be a single 1mm2 cable (though 1.5mm2 seems to be more commonly used).

You aren't expected to be running everything at capacity at the same time. Just because a circuit can supply a given amount of current doesn't mean that it will be doing. As long as the current draw at any point doesn't exceed the capacity of the fuse or breaker protecting it, it won't trip.

You mention 3A and 5A "plugs". This is the current above which the fuse in the plug should blow (at some point - fuses have different "trip times" for different overcurrent ratings). The appliance supplied by these won't normally draw the current marked on the fuse. For example if you had a 1W appliance, it would be likely to have a 3A fuse (the smallest common fuse for a UK 3 pin plug), but would only actually draw around 0.004A. You could technically therefore connect around 3000 of these 1W appliances through one 13A fused plug without a problem (practically however, you might find it more of a challenge...).

The important number to be considering is the total power draw (wattage) of the appliances in use at any one time, not the total of the fuse ratings.

In relation to you aquarium, as long as you don't overload the ring main, you'll not have a problem. If you total up the wattage of all your equipment, you can run approximately 3000W through one 3 pin plug. (Note though that it's common for double sockets to be rated for less than 26A, but it's not often marked on them). Your 32A ring main is therefore capable of supplying a bit over 7000W.

In the worst case, you could always have an electrician install a dedicated circuit to supply the aquarium equipment.

[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.