No electric expert but you don't need a 5V voltage regulator when you use the 5V from the Arduino it is allready regulated.
You state you use the WIZ5100 library, is it compatible with the WIZ810MJ ?? Have you checked the datasheets?
This module requires 3.3V and yes the module is compatible with the standard ethernet library.
@simp you need to connect the RESET line to a digital output pin with a 10k resistor. Set the output HIGH for normal operation and you can toggle it to reset the module in code.