Which smart home system is suitable for our new building?

What do you want to secure against what?

If you want to use VLAN, then you need a router that can do that. Then you are out of the AVM universe, as they at most offer an additional guest network. That means dealing with semi-professional routers. These require some time and knowledge to set up properly. I have stayed with the Fritzbox as a router because I am too lazy to get into it. For myself, I have decided to only include devices in the network that I "trust," for example, no cheap Chinese junk. Additionally, you can also forbid these devices from phoning home.

This will be the speed of your internet connection. But there are also scenarios where more in the local network is great, if the laptop is only connected via Wi-Fi.

Those definitely exist, but for that, the laptop is then connected via LAN, just like the NAS.
I don't do data backups with the laptop on Wi-Fi.
Wi-Fi is mainly needed for TVs, smartphones, tablets.
And they have to handle larger amounts of data than my internet connection can.

I dare to doubt that.

In this respect, I think you can relax about the access points. As I said, we're talking about a single-family house where 60-100m² per floor need to be covered.
Not an industrial hall or a hotel with 500 rooms.

I have one more question about POE

I would install a POE switch in the basement for distribution to the rooms

Can I then connect a small POE switch in the rooms (if needed) that does not require an extra power cable?

Can devices that do not require POE be connected to a POE switch or would they be damaged?

Can I then (if needed) connect a small poe switch in the rooms that does not require an extra power cable?

You want to split the POE again? No, it’s not that simple. You will need another POE switch or injector at that point.

Can devices that do not require poe be connected to a poe switch or would they get damaged?

That’s of course possible. Normally, the switches themselves detect which devices require POE and which do not.

Just buy an 8-24 port POE switch (depending on how many ports you need). You connect everything you have to it and that’s it.

I will install a Cat Duplex everywhere.

But if I need more than 2 LAN connections somewhere, e.g. behind the TV, I wanted to install a small POE switch there (4 or 5 ports).

For the 2nd POE switch behind the TV, I then need an injector because the 1st POE switch in the basement cannot provide enough power to operate another 4-5 port POE switch.

I will install Cat Duplex everywhere.

I assume that.

But if I need more than 2 LAN connections somewhere, for example behind the TV, then I wanted to install a small POE switch there (4 or 5 ports)

Thinking already now about being dependent on crutches again later is counterproductive. Think three times (better ten times) about where you will definitely need LAN now and in the next 10+ years. Especially in the media corner, I would probably go for 4 connections from the start. Then rather save somewhere else.

For the second POE switch behind the TV I then need an injector because the first POE switch in the basement cannot provide enough power to operate another 4-5 port POE switch?

No, an injector supplies the specific line with the necessary power, but it’s also a crutch. A second POE switch would of course provide the required power for the available connections. BTW. Most of the things that currently need POE are access points and IP cameras, possibly also phone systems. So behind the TV you will most likely not need POE. But plain LAN. Unless, of course, you still intend to hide a POE access point behind the TV.