graphics card never improves the quality of 1080p pictures rather the quality depends on the picture itself...Better the quality of picture more faster/powerful would be the card required...But still even a card as powerful as a 5450 would be enough to run the best quality motion pictures
Would the 5450 be noticeably better than the 3200?
Also, for some gaming over the HDMI connection does the card make a difference then? This computer is mostly an HTPC, however, I'd like to be able to use it as an extra gaming machine. For good gaming graphics do you have to use DVI or will HDMI work just as well?
The 5450 has close to no gaming capabilities, the 5670 or higher would be better suited for your gaming experience. What screen resolution and texture settings do you expect/want to play at?
In the link in my first post Toms rates the integrated HD 4200 as an 80, so since the 3200 came before it I would assume that it scores less so the 5450 would be the absolute lowest performing card that gives the maximum performance for your HTPC needs.
Certainly 5450 is better than 3200...It makes no difference whether you game with DVI or HDMI as both are inter convertible via an adapter...The main difference lies with the graphics card itself..5450 is certainly more capable than a 3200 but still its one of the worst option available for gaming...
No it won't give you a better 2D picture. An HDMI signal is digital so you don't have to worry as much about em interference as you would with an analog connection like a VGA connection. Anyway, a dedicated graphics card will give you better 3D and GPU accelerated performance, but it won't improve the picture quality of your desktop or watching videos .
Video cards do affect the overall video quality output, and yes in general, more powerful cards can perform better than less powerful cards due to functions that may be limited or disabled on less powerful cards. There is a point of dimishing return though or no return at all. For example. I would guess that a HD 5770 will offer no better playback quality than the HD 5670 or HD 5650.
One thing to remember is at a certain point "quality" becomes subjective as to what actually looks better.
Below is a link to video quality of some video cards that most people would put in a HTPC. The cards that scored the highest are the HD 46xx series and the 9600GT:
5670 would be a good choice, however for 1080p with AA on you'd want to go the 5770 route, which is a bit more expensive. But if you are using it for light gaming I couldn't see a reason why a 5670 wouldn't be adequate enough to run the games on lower settings.