Black Screen of Death

[Alexios]

Well, for me it seems that your graphics card is almost incapable of running this heavy game, as it clearly was too heavy for my GTX260, which is actually significantly better than your cheap graphics card.
Also you can try disabling sleep mode in your system.
And overall, I think playing on a laptop isn't actually a good idea at all. Most of them are completely incapable of handling big heavy games such as this one.
Considering that you have a laptop, I would also suggest you check if it overheats when you have this issue next time. This is also a common issue with virtually all laptops.

[Elkwood]

I suggest you either do 1 of 2 things. If you good with laptops crack that baby open and apply some new thermal paste to that GPU, the black screen is caused by your video card overheating. Also ensure that you drop all your XIV settings to the lowest possible option and see how the game responds, Turn off AO, AA, Depth of field etc. The 2nd option is for you to buy a fan to put underneath the laptop to get it some airflow.

Display mode saying 32 bit just means it is displaying 32 bit graphics for a desktop, you may have a 64 bit OS but I guarantee you XIV is not taking full advantage of it because it was designed for 32 bit OS, so the game itself will also run in x86 mode.