I'd be really surprised if this operates, at least with any reliability. You desperately need a power plane and a ground plane with components like these.
For instance the ENC28J60SS is a current mode driver, those traces from it to the pulse transformer to the jack are going to be carrying high currents switching at high speeds with no reasonable supply or ground path for them. I'd also be concerned with the crystal as its tuning capacitors have a very long high inductance path to ground. I would guess that this design will emit a ton of EMI, possibly enough to interfere with any nearby electronics.
The layout for the switcher is also huge, i would expect you to have serious amounts of noise and supply ringing throughout this design coming from the switcher and the ethernet components.
You also mentioned POE but there is no POE controller on this board. If your planning to just plug into a POE switch there is a negotiation procedure to request power (in various levels). The only way you could power this off the ethernet port as is would be to find a mid run injector that just dumped voltage on the unused pairs in 10/100 Base-T. If you plugged this into a gigabit switch that could cause major problems. Those pairs also aren't center tapped as they should be for POE. Dumping pins 7/8 directly to ground could cause major problems for both this device and the switch.
There is a lot i could go into but 90% of it is a result of not having power/ground planes, clear that up first.
Also the mid point of R5 and R6 should not be connected to the center tap of the pulse transformer and the jack side of the Output transformer should have a small, high voltage cap to chassis ground at its center tap
Best Answer
There's an existing Arduino library that uses the module's UART interface, rather than the more common SPI. It provides basic methods for reading the UID of a card and R/W operations on blocks. You could easily port that library to your favourite PC programming language, connect the module to your PC probably through some USB-UART adapter and get to work.
If your PC has an actual TIA-232 serial port and you want to use that to talk to the module, you'll need a driver like the MAX232 to sit between the module's TTL UART and the PC serial port.
Use this as a guide:
All GNDs must be connected together.