Unless you are very price sensitive, using USB Power Delivery ICs is the norm now for most devices, but PD is different than POE. PD is just loose tolerance resistors on USB.
Incorrect. https://en.wikipedia.org/wiki/USB_hardware#USB_Power_Deliver... is a good start about the subject: "PD-aware devices implement a flexible power management scheme by interfacing with the power source through a bidirectional data channel and requesting a certain level of electrical power <...>".
No, it's not. You can do very basic selection with resistors, but you can't get above 5V (or more than a couple of amps IIRC) without using the actual PD communication protocol.