How to implement PoE with WIZnet W5500?
The diode bridge looks to be reverse polarity protection when using the jack to power your circuit. The jacks with these diodes in it can not be used to provide power to a remote device.

Additionally the W5500 seems to require independent center taps to match the reference designs which limits the selection further. We found the following to fit the requirements but it is so far untested:

https://www.digikey.com/product-detail/en/abracon-llc/ARJ-199/535-13149-ND/5430434

As far as power - you should use a regulated supply. 48 is part of the POE "standard" but some devices do not conform to that. You would need to figure out what device you plan to use if you plan to use a lower voltage. That said, lower voltage means higher current for the same power output. 

The actual standard involves some sensing and ability to adjust or remove power under certain circumstances but will require more circuitry to implement.