1. The common one is to use Bluetooth or WiFi for wireless. Bluetooth you can try Arduino BT joystick Free or BlueTerm. WiFi you can try WiFly Remote. You must develop something around the apps and Arduino, develop code for it. Or at least find tutorial regarding it.
2. Obviously, Arduino UNO or other compatible controller, LEDs, resistors, Bluetooth module, WiFi module, shield, etc. The list goes on. We can only give example, all projects are different when you go detailed, and only by doing it you will know exactly what is needed.
3. Well, that depends on the wireless technologies, and module. Bluetooth is point to point, no modem is needed. WiFi, depends, If you are using XBee-WiFi, it support soft AP, no router is needed.
You will have to do some study, I am sure there are more questions. Answering those question is copying the text book about wireless and electronics and paste here.