Yes, the power supply unit is included

This switch runs at 1Gbps. Check that the cable is at least cat.6 and the network card where you attach the other end of the cable must also be at least 1Gbps.

I probably don't understand the question correctly. Should the TV box receive an IP from the switch or from the router? Or do you really mean that the switch should receive an IP from the router? Depending on the device (router and switch), you can set whether the IP is to be obtained via DHCP mode or manually defined as a fixed IP. If DHCP is selected, an IP is assigned in the defined DHCP IP range, and the remote device must act as a DHCP server accordingly. The TV boxes usually cannot be changed/adjusted. With my TV box (not from UPC), I also wanted to assign my own fixed IP, but this is not possible/can not be set. Have you set up your router as a DHCP server ("IP provider")?

Need the part as a flat distributor to connect all rooms. 3 x Swisscom TV Box and 2 x PC. I have never had a problem so far. (PS: The flat is connected via fibre optics).

Sure... whether that makes sense in a specific case is another question. To answer that, you would have to describe in more detail what you are planning.