Brushless will last much longer, are more efficient, and offer a much wider range of ESC's.
Arduino Uno is more than sufficient for basic multicopters.
Cannot say about the shields. HobbyKing offers MultiWii controller boards which incorporate Gyro's, magnetometer, Accellerometer, and will support other modules, such as GPS, for about $30 US (but shipping is really costly).
Before you start pulling parts together, consider:
Foamie may not take the stress without additional structural support (wood, rigid plastic, or metal).
In my limited experience, the lighter the bird, the harder it is to stabilize. design heavy and it will be easier to add things like camera and gimble later.
Also, you might consider starting with a tricopter design first, much, much, easier to tune and balance (IMHO).
Good Luck and enjoy!