Technical room / Router / Access Points / Switches

You are already using 15 ports. Just get a 24-port switch.
The most sensible option is to lead the LAN installation cables directly into the network cabinet. There, terminate them on a patch panel and connect to the switch using patch cables.
If you don’t want to do that, you could mount keystone modules on the installation cables and connect directly to the switch with patch cables.

Only power, fiber optic, and one LAN cable to the switch go into the Fritz!Box.
Fritz!Box 7590 is the wrong box because there is no fiber optic connection! You need the 5530 Fiber or the smaller model. Both work for your purpose.

I would also take a PoE switch.

Regarding APs. Sure, you also have Wi-Fi 3 meters away. But if there is a wall or similar in between, you don’t get fast 5GHz Wi-Fi but slower 2.4GHz Wi-Fi. For normal use, that is still sufficient.
The access point you mentioned fits. Or a slightly cheaper option I suggest.

I would do LAN in the garage and garden right away. Especially if it’s not plastered yet, etc.
And then you’ll already need a 24-port switch ;). We’re not even talking about cameras and such.

One more question regarding the patch panel:

The electrician says there will be an 18-port patch panel included. That's where all the LAN cables are brought together and they are then inside the electrical cabinet.

What does that mean for me?? Do I have to unplug everything and then separately connect each cable individually to the switch?

All the better. So directly patch cables from the patch panel to the switch. However, the cleaner and proper way would be to run installation cables all the way to the network cabinet and have the patch panel there as well.

All the better. So directly patch cables from patch panel to switch. However, it would be cleaner and more correct to run installation cables to the network cabinet and also have the patch panel there.

So I basically have double the effort or double the material costs because I have to run another LAN cable from each LAN port on the patch panel to the switch?

In the end, the effort is always the same, right? I definitely need an additional LAN cable from the patch panel to the switch? So there are 14 individual LAN cables running from the patch panel to the switch, or how should I understand that?

And one more question:

What is ultimately the benefit of the patch panel if I have to "extend" all the cables anyway? Whether it is in the network cabinet or in the electrical cabinet doesn't matter, right? I can just lay all the cables individually directly to the switch?

With the patch panel, you can patch. The installation cables are quite stiff. Theoretically, you can also put connectors on installation cables and connect them. But this is ABSOLUTELY not recommended. There are also no double costs. Patch cables don't cost much money. Just make sure that either the patch panel goes directly into the network cabinet or the network cabinet is nearby. Attached is how it looked in my setup. You would only have a patch panel and a switch.

You can patch through the patch panel. The installation cables are quite stiff. Theoretically, you can also put connectors on installation cables and connect them. But this is ABSOLUTELY not recommended. There are no double costs either. Patch cables don’t cost much money. Just make sure that either the patch panel is installed directly in the network cabinet or that the network cabinet is nearby. Attached is how my setup looked. You would just have one patch panel and one switch.

Understood! Thank you very much !!

Is it ultimately also possible afterwards for the electrician to easily mount the patch panel into the network cabinet? Do the "rails" etc. always fit universally? Does it help to talk to the electrician to have him not mount it but leave it loose somehow, so I can take a look at it and then mount it myself afterwards?

Is there a 24-port POE switch that you can recommend and possibly a suitable access point? Affordable price :)

And also a suitable network cabinet? Can you just buy any 19" cabinet? Is there anything to watch out for? Especially now regarding the electrician’s patch panel?

Ultimately, the effort is always the same, right? I definitely need an additional LAN cable from the patch panel to the switch? So, 14 individual LAN cables going from the patch panel to the switch, or how should I understand this?

Yes, but here you get the set of short colored cables for < 15 EUR and everything is neat and clear.

And one more question:

What is the actual benefit of the patch panel if I have to "extend" all the cables anyway? Whether it hangs in the network cabinet or in the electrical cabinet doesn’t matter, right? Can't I just run all the cables individually directly to the switch?

I doubt that the cables end up in the electrical cabinet; they are probably led out beside it, right? So here on a patch panel. If the electrician is doing the termination anyway, get him a suitable cabinet (around 100 EUR) and a corresponding patch panel so he can terminate the cables directly on it. Then a switch (+ Speedport Smart 4 plus according to my recommendation) and everything is good in connection with the mentioned APs.