When I use Illustrator CS5 on another machine, all objects X and Y are relative to their artboard, not the overall pixel grid (by pixel grid I mean the entire screen). How can I change this so the X and Y values are not relative to the object's artboard?
You can reset your coordinate system to any point on the drawing surface. Just turn on your rulers and drag the vertex to the point where you want your new XY intersection to be.
Keep in mind that this will not move existing art, it will just change the point that the coordinates are relative to. For example, if you have a default letter-size artboard with an object at 0,0 - it will be located at 7795,7885 after moving your coordinates the bottom-left corner of the drawing surface.
In CS6 right click on the ruler and select Change to Global Rulers. Not sure if this applies to CS5.
If you require pixel-aligned placement, you will need to set the rulers and measurements to pixels and also use the pixel preview setting under View in the menu bar to ensure what you see on screen at 100% is what you are going to get when you export to a raster format.
The artboard is not necessarily a hard limit set by the application. You can set the artboard to whatever pixel dimensions you require and the work with Illustrator as you otherwise normally would using the artboard as a guide. You can even set the origin of the rulers to be anywhere in or out of the artboard or simply turn them off is you find them distracting.
View -> Rulers. Right-click your ruler and Choose 'Artboard Rulers'.