Most likely not with the graphics card you're planning on buying. RAM is usually about $30-50 so it would be better to up your graphics card budget $30-50. You'll see better FPS with a better graphics card. That is what games rely on the most.
Not really. BF3 can use more RAM than 4gb but at the settings that you are using it is highly unlikely that you will see any problems with it. I played through the campaign on Ultra with 4gb of 1333. No problems
To your statement about your DDR and GDDR matching. This is completely false. They are independent of each other. Speeds and even types will not have any effect on each other.
Edit: claptonman is right. The money that you would spend on RAM would be better spent on a better GPU.
there would be no effect at all of getting more ram or faster ram. So if you can spend the 30 or so the ram would cost to get a 6770 we would recommend you do. and since you want to play BF3, if you can spend the extra 30 more to get a 550ti, i think that would be smart, as BF3 like nvidia cards.