Which router for our new building?

Does it also have to be a (PoE) switch from the same manufacturer for this? Does it bring any advantages?

Only if there is a common management from the manufacturer, e.g. like with Ubiquiti.

If I have only 3 PoE access points, can I connect them to a "small" PoE switch and use a non-PoE switch for the remaining LAN ports? Is the connection between the switches foolproof?

It will work. The uplink from the small switch is then the bottleneck for all APs together. You have to consider for yourself how heavily you load the APs or whether a Gbit uplink could become a bottleneck.

What happens if a 24-port PoE switch – apart from costs and possible noise disturbance – is also used for non-PoE devices? Can something get damaged on the receiving device?

Nothing happens, the devices negotiate among themselves whether they want to draw power or not.

I am always a fan of central switches if cable lengths and installation allow. In general, the division is not a problem as long as the uplinks match the intended usage.

[*]Does it have to be a (PoE) switch from the same manufacturer? Does that bring any advantages?

Not necessarily, but it can make management easier.
I use different manufacturers.

[*]If I only have 3 PoE access points, can I connect them to a "small" PoE switch and use a non-PoE switch for the rest of the LAN ports? Is connecting the switches foolproof?

Yes, or you can work with injectors.
Otherwise, the power consumption is high with a large PoE switch.

[*]What happens if, for example, a 24-port PoE switch — aside from cost and possible noise disturbance — is also used for non-PoE devices? Can something get damaged on the receiving device?

That is not a problem.
But see the power consumption of the switch.

Does it also have to be a (PoE) switch from the same manufacturer for this? Does it bring any advantages?

No, it does not have to be from the same manufacturer. The POE standard must fit, so that enough power per port is available and the POE budget must match all devices. There can be advantages if everything is from one series and switch + APs can be managed via one interface. But I can't say anything about Netgear here.

If I only have 3 PoE access points, can I connect them to a "small" PoE switch and use a non-PoE switch for the remaining LAN ports? Is the connection between the switches foolproof?

You can do that. Depending on the application, the connection between the two switches can be a bottleneck. However, I don't see major problems in the home network area. Otherwise, there are also devices that ensure a connection with higher transmission rates.

Can something be damaged on the receiving device?

No, a non-PoE device cannot be harmed.

thank you all!
just a quick note about power consumption. If I only have 3 PoE-enabled devices on the PoE switch, then I only have 3 power consumers, right? So it shouldn't matter whether the 24-port switch is PoE-enabled or I have an 8-port PoE switch with just 3 devices? Or am I mistaken? This is regardless of the PoE standard.

Thank you all very much!
Just a quick note on power consumption. If I only have 3 PoE-capable devices on the PoE switch, then I only have 3 power consumers, right? It should then be irrelevant whether the 24-port switch is PoE-capable or I have an 8-port PoE switch with just 3 devices? Or am I mistaken? Now, independent of the PoE standard.

The problem is that switches that can deliver a lot of power often also have a high base consumption.
So high efficiency is only achieved at around 90% utilization or similar.

I pray to God that you will still be this active here in March when things get serious with the wiring on our end. :)