The 670 should fit in that case, the GTX 670 is 9.5 inches long and I've never seen a ATX case with a bay below 10 inches, so that'll be fine I think. also, the PSU will be more than enough to power your system and it's a good quality brand aswell. You should be good to go.
There is another thread where someone asked about how long a card would fit within your case you have 11 inches of clearance. The gigabyte version is very long its on a 680 PCB but should be just fine with your case also keep in mind that you also have extended room because of how your pci slots work anyhow. By the way I have the same card as you plan to buy hope you enjoy it if you purchase it.
The gigabyte 670 is one of the stronger cards to purchase. It does use a larger PCB and also requires a 6+2 pin connection which differs from the normal two 6 pin connections of the other cards. If I were to purchase a card again I would just get the normal ASUS 670 DirectCUII. Both are very quiet cards.
As for which card, if ya have a gigabyte board I'd lean towards the Giga GFX card as it makes talking to tech support if ya have a problem a little easier.....1st thing they learn in TS101 is "get the customer off the phone by blaming the other vendor".
My fav 670 is the Asus DCII......heres some reviews:
I woulda bought it if you could find the damn thing at launch, got a Gigabyte instead and I am happy, there is a 2 fan model of the Gigabyte 670 I believe that is a little shorter, the 3 fan one is pretty long.