Update. I Went with the HP Pavilion system.
So dimensions: 16.85"(L) x 6.97"(W) x 15.32"(H/D) are what I'll be working with here. What is most important in determining what a card can fit? Length, height, width? And in what order?
The most important thing is the length, in most cases it is the only thing that matters.
The height of most GPUS are identical, with the sole exceptions of smaller cases that only supports half-height cards. Meanwhile, the width of the card matters only when you have expansion cards (Wifi, Audio, etc.) in the slot next to the GPU, as some GPUS (like the HD5770) takes up 2 slots.
In your case you will be fine, the Mobo has a decent layout and your case should have enough space, just get a new PSU as the 250W in that system is very weak.