Telephone connection / LAN wiring

Besides the LAN wiring, we are currently tending to also lay a telephone cable in two places in the house (kitchen + office).

Nonsense... lay LAN everywhere... then you can also use it for telephone as needed... it doesn't work the other way around... so you might end up with cables lying around that are useless.

If I were to place the router, for example, in the office and connect it via the telephone cable, I could connect the LAN cable at the LAN port, which runs back to the house connection room, and there supply the right side of the house via LAN through a patch panel.

Just lay 1-2 extra LAN cables into the office and that's it.

Are there now phones that can be connected via WLAN or do I still need a telephone cable from the router to the phone's location?

There have been DECT phones for decades and these work very well inside and around the house. Besides your WLAN.

Besides the LAN wiring, we are currently inclined to also install a telephone cable in two places in the house (kitchen + office).

Nobody installs telephone cables anymore today. Nowadays, a "structured cabling" is chosen, meaning network cables are laid everywhere "fully" (i.e., with all eight wires). Whether an analog telephone later uses only two of them doesn’t hurt. IP phones are technically data devices, even though they provide a "voice service" (and use all wires for Gigabit).

Are there now phones that can be connected via WLAN or do I still need a telephone cable from the router to the place where the phone is set up?

Connecting phones via WLAN only makes sense where they are mobile phones, i.e., where you want to use the mobile phone at home. Cordless home phones work better with DECT. The DECT base can also be an IP DECT base (meaning that between the base and handset is DECT or cat.iq over the radio link, and between the base and the telephone system is IP). This IP DECT base can also access the PBX / router via WLAN.

Where a Fritzbox is poorly positioned for the use of directly registered handsets / Fritzfons, an "ISDN" DECT base at the FON S0 is usually used.