Powering Arduino and LED strips from one adapter
Powering an Arduino and LED strips is no different to powering your TV, satellite receiver, DVD player, etc all from one socket in the wall.

Connect all the grounds (-) together, and connect all the powers (+) together.

All you need to worry about is that the total current draw of all the devices combined doesn't exceed the amount of current that the power supply can provide (oh, and that they all operate at the same voltage, of course).