currently the video is radeon 3450 hd card, but there's an hdmi slot on the pc from like built in graphics 3200 series i think.
my main pc has a 8800gt 512mb dvi only.
Im not sure which video card you are using now, I am not familiar with nVidia products, but with ATI, set your resolution to 720p in the catalyst control center and adjust the overscan if needed to fill the screen. Small text will not look 100% crisp because the image is still being scaled, but games and HD (720p) video will look great.
As for 1080p and 1080i being practically identical, while in theory this maybe true, but on a 1080i HDTV set that does not support true 1080p there is a huge difference. The TV will scale the 1080i image to the native resolution of the panel, which in most cases is 1366x768 or 1360x768.
Running 720p to your panel is the best resolution over a digital connection (DVI or HDMI) since the least amount of scaling will be done with that method. If you want to run the at the native resolution you will need to run an analog dsub (vga) connection to your tv, if it supports it and set your resolution to 1366x768 or whatever the native resolution might be.