First off I'd like to say, Welcome!
I have an 06 WRX and I actually moved the pocket to the top and put my 700IDT in the bottom. But since you are keeping your radio, that probably wouldn't work so well for you. If you put the monitor in the top one, you'll not only lose the cupholder but also the air vents will be completely blocked.
As far as fitment goes, I'm not sure about the newer 700IDT, but the older doesn't fit width-wise into the dash, I had to buy one from a junk yard and use a dremel on it to widen it(It looks pretty clean). Make sure you check the dimensions of the 700IDT though.
Now for cases, your main problem is gonna be the height, meaning you can't have any PCI cards most likely. I don't know of any ATX cases that would be thin enough. But you could always make a case out of Acrylic or Lexan. I just use a mini ATX case with an ITX mobo and its sits in my trunk right beside the sub.
And for the heat, I don't see that being an issue at all, if you have a fan in your case it'll be fine.I would be much more concerned about the cold if I were you