Number of network sockets; how to plan Wi-Fi in the new building?

APs do not belong in the hallway area but rather where the most speed is needed.

However, the distances are shorter for most than for you. The position in the hallway is sufficient for the entire upper floor. The other one is at the TV on the ground floor. What is definitely not sufficient is a single access point on the upper floor with a reinforced concrete ceiling and certainly not a single Fritzbox in the basement.

It is not just the distance. Walls play a big role at 5ghz...

In most houses, a well-positioned access point per floor is sufficient. We implemented this with a friend 3 weeks ago. The Fritzbox is only used as a modem. Additionally, there is a Unifi USG and POE switch. Connected to it are also a NAS and 5 cameras. In total, there are 9 POE devices (3 APs, 5 cameras, 1 Cloudkey). This way, he has all options with Vlans, etc., and it is centrally manageable. Everything fits comfortably in a small 19" server cabinet (wall-mounted). Positions are the office in the basement, the living/dining room on the ground floor, and the upper floor between the bedrooms (no solid walls). Everything that is stationary is connected via LAN. This way, you have a fast, stable, and future-proof network. Better to plan one more connection so that you can expand if needed. Mesh, repeaters, Dlan, etc. have no place in a new building.

"APs do not belong in the corridor area but where the most speed is needed." if this is the optimization requirement. However, if one prefers to get by with the minimum number of devices for complete network coverage, in many cases the corridor variant is simply the most cost-effective.

Do I need a Cloudkey? My brother-in-law and I are unsure about 2 APs.

No, you do not need a Cloudkey. You can set up the APs as standalone or install the controller software on your computer. As soon as the controller is offline, some functions like VLANs no longer work and there is no monitoring of your network. If you have a NAS (Synology or Qnap), you can also install the controller on it. Since it usually runs 24/7, the controller is always online. The Cloudkey is a small computer on which the controller software runs. It requires little power and if you want to use some functions, the controller should always be online. This is good if you do not run Docker or a virtual machine.