No, you cannot operate a UniFi access point via PoE on this switch. The switch only has 1 PoE port to the outside, not all ports. It itself is operated via an external power supply (USB or PoE input) and then only supplies devices with up to 6.4 W (802.3af) via this one port. The other 4 ports are pure data ports without PoE output. For an AP, you typically need a PoE-capable switch with more PoE power.
You can use the switch as a simple distributor as it supports Layer 2 functions and does not require any special configuration to function as an unmanaged switch. However, it also offers cloud management via the UniFi Network App, which is optional. There should be no problems if you use it without a controller, as long as you don't need advanced management features.
As the previous speaker has already written, a single unifie switch makes little sense. In your case, a normal unmanaged switch will suffice. There is only a loss insofar as all devices share the bandwidth. In practice, you usually don't notice much of this because not all devices need bandwidth at the same time.
Yes, you can use this switch standalone without a controller, but it's designed to be managed via the cloud with the UniFi Network app. This means that to take full advantage of its advanced features, it's best to integrate it with a UniFi controller. However, it can operate as a standalone switch for simpler networks.
Your considerations are basically correct, but there are a few things to bear in mind:
1. fibre with modem in bridge mode: the modem should be able to be connected to port 1 of the switch without any problems.
2. connection Ubiquiti No. 1 to modem: Port 2 could be used for Ubiquiti No. 1, but you must ensure that the PoE injector is inserted correctly, as the switch itself does not output PoE, but can only receive it.
3. connection mini PC to port 3: No problem as the switch supports 2.5Gbps.
4. connection house LAN on the ground floor and on the 2nd floor: Port 4 could be used for the house network on the ground floor. For the 2nd floor, you could use another PoE injector to feed Ubiquiti No. 2.
However, the switch only has one PoE input, no PoE outputs, which means that you need a separate PoE injector to supply Ubiquiti No. 2. So your ports are fully occupied, but the configuration should work if you use the PoE injector correctly.