For large LED installations, I recommend using a "USB native" microcontroller like the Teensy 2.0, Arduino Leonardo or Micro. The serial connection on the Uno isn't fast enough. For any of these alternate microcontrollers, you'll need to adapt the wiring from what's in the tutorial to each board's unique SPI pins.
That's an ideal power supply, yep.
Six strands of 25 LEDs, yes. But for long runs like this, you can't just power it from one end...you'll need to "fan out" the power to multiple connection points (I would suggest three, using 18 gauge wire). See the Advanced Topics page
of the Adalight tutorial.