1: if you are looking for a good case, the haf xm (same as the haf x but just a tad smaller) or the corsair carbide 500R are pretty much my picks
2: id get the gigabyte gtx670 or the msi power edition 670 instead of the asus. they cool better than the asus and for the msi one, it has overvolting support. no other 670 has it
3: the video cards i recommend are 10-11 inches. the asus is 10.75
4:the power supply you have currently can handle the card fine. you might want to look into something like the xfx xxx 750w power supply if you are going to do some SLI. its possible on the gs600w but i dont really recommend it
To answer your questions:
1) GTX670 of any kind is not power hungry, and should run on a good 500w psu. Your Corsair is a good one, and could run even a GTX680.
2) It should fit, but I can't confirm that from experience.
3) Functionally, a modular psu does nothing but save the space for the few power leads that you are not actually using.
What don't you like about your current case, and what does the HAF-X do for you?
I might suggest you look into the Silverstone TJ-08E if you want a smaller case.
I have the same M-ATX motherboard in the TJ-08E along with a GTX680 and a full height cpu cooler.
The thing is silent, the front 180mm fan provides plenty of silent cooling. http://www.newegg.com/Product/Product.aspx?Item=N82E168...
My current case is small, that is why i want the HAF X, it has plenty of room, cooling options and good cable management. I can't fit the H100 on my current case and with the HAF X i can fit it and still use 2x 120/200mm fan on top