I'd recommend the HDHomeRun. It's a network based dual ATSC/QAM tuner. I don't have one yet, but I will soon. It's well supported in Linux, and the best part is that it doesn't take up any PCI slots in your PC.
Yes your box will upconvert DVDs. If you set your video card to output 1920x1080 then DVD playback will occur at that resolution.