How to handle multiple GND, and +5v vs VCC
When you see ground or VCC connections they symbolize a common net. That is, it's a shorthand way to show all the pins that need to be connected to those rails without actually having to draw a ground or power wire on the schematic.

As you surmise, all the grounds, and +5Vs and Vccs, are indeed connected to their respective "together".

In this case Vcc is actually mislabelled, and badly so in my opinion. 

Your Vcc shown on the schematic should actually be labeled as "RAW".  It's simply the rectified and smoothed 9V AC voltage. The +5V signal is generated by the Arduido, and should more rightly be labelled Vcc. 

Whether the arduino can actually supply enough current for all the things attached to that pin in unknown.

As for where to get 15V and 9V AC, you would need suitable transformers. If you want to use a laptop charger to get your RAW voltage, you would need to down regulate it to within range of the ARDUINO RAW input voltage range.