Arduino for delivering a WoL signal over ethernet, when receiving a specific command over WiFi
The Arduino might not be a choice for you. The reason why microcontroller boards(like Arduino) become popular is the simplicity and the easiness of control peripherals through GPIO pins, like output PWM signal to control motors, reading sensors through SPI or I2C, reading analog voltage, and etc. Most microcontrollers on Arduino boards itself, however, is not designed for networking features and does not have even enough performance for it. This means that it would cost you much more money on additional networking devices like WIFI shield, Ethernet shield and microprocessor for networking (Arduino Yun).

So, if you are looking for just a pure networking device like what you have in mind, I would highly recommend you Raspberry Pi(2 Model B for $39 and 1 Model B+ for $30), which has enough capability of networking itself. All you need is a USB WIFI dongle($9) with the Raspberry Pi, I guess. I am pretty sure you won't have a cheaper option using Arduino.

You can use the raspberry Pi as a web server, a NAS, a media server along with the features you mentioned.