PCIE 2.1 is backward compatible to 1.0, so no problems here.
Good observation on dimensions of card.
This is your PC with open side panel
This one looks like the same or not? Below
The question is - is dual bracket card will fit, I can see space for dual bracket (will occupy PCI slot above PCIE slot), just the length of the card
And now I found comparison GT630 vs HD6670
http://gpuboss.com/gpus/Radeon-HD-6670-vs-GeForce-GT-630
HD6670 wins by numbers.
Remembering $80 parameter let me give you a few more 6670 cards with better dimensions
First is dual bracket SAPPHIRE 100326-2GL Radeon HD 6670 2GB 128-bit DDR3
http://www.newegg.com/Product/Product.aspx?Item=N82E16814102992
Card Dimensions 6.46" x 4.21" x 1.57" I think that length of the card before heatsink shroud must not exceed 7.5", so just in case you can measure it to confirm, in this case this card fit.
A few more, they just don't look so "masculine" as the other ones
Next is perfect fit
ASUS HD6670-2GD3 Radeon HD 6670 2GB 128-bit DDR3
http://www.newegg.com/Product/Product.aspx?Item=N82E16814121645
This one is only 1GB card and as such is very skinny
XFX HD-667X-ZHF3 Radeon HD 6670 1GB 128-bit DDR3
http://www.newegg.com/Product/Product.aspx?Item=N82E16814150542
So, if you not sure, last one will be perfect, if you can spend a few buck more Asus will be better because 2GB will give you more resolution and better support for multiple screens (if needed), same goes for Sapphire.
Anything else I can help you with?
P.S. If you find any of above posts helpful and ready to close this thread please pic up as solution most useful post.