Controlling servos with Arduino is pretty easy since a "servos library" is included in Arduino. Here is a good examples
to start with.
Using this servo library, you can control up to 8 servos. If you ever want to control more than 8 servos, you will have to use a separate servo controller. (Pololu and Lynxmotion provide nice servo controllers).
All servos can be powered from 4.8 Volt to 6 Volt. The higher the voltage the more torque on the output shaft. The voltage vs. torque values should be given in the servo datasheet so you can have an idea on what to expect.
So, to answer your questions:
1) Any Arduino board will do the job. You need 1 analog output pin per servo, so 8 analog output pins to control 8 servos. The Arduino Uno and the Arduino Nano have 12 analog output pins each and the Arduino Mega has a lot more.
You can get a very cheap Arduino Nano clone "made in china" on Ebay (~10€). Note that when you by a clone, no money goes to the official Arduino project :-(
2) You can run them all at 5V from the board to avoid using an external power source. Try to find how much current you need to power 8 servos (each servos are different have a look at the datasheet) and if you ever have to use an external power source, just connect it to the red wire on the servos. (or maybe you can use the VIN pin, I'm not sure)
3) In that example, the servo is powered at 5V from the Arduino. No need for voltage regulation here.
Hope this help.