I'm not really the person to ask regarding low end video cards and things like the best one for media playback and post processing etc. but as far as I know yes all modern cards can do 1080p. The only thing I can base this off of though is the fact that I used to have a 4350, which was basically the least powerful and cheapest card available 18 months ago, and it could almost do 1080p and could do 720p without a problem, so I'd imagine anything stronger than a 4350 would be able to do it.
Let me try and find you a useful link regarding good cards for media playback.
EDIT: Here give this
a read, it's a quick look at the 6450. In it they mention that while they haven't run it through the full gamut of their media test they expect good results from it. One downside of it is because it's brand spanking new it will be a little more expensive for what you get. In the article they link to their previous review of the gt 430 which has been out for a little while so it's price should have stabilized, and you can read that review to determine if it meets your criteria.Edited by juano - 4/16/11 at 10:25pm