If your motherboard has a hdmi output and the monitor has hdmi input then yes. However i would try both dvi and hdmi. For my monitor i found that dvi gave me more consistent picture quality than hdmi.
Depends on the TV's video processor. My Samsung 42inch lcd limits my frames when running 1080p via HDMI, very noticeable when moving your mouse cursor around.
Enabling your TV's game mode can alleviate this at the cost of some picture quality, although it just makes your pc games look something akin to xbox 360 graphics. Seeing as i only use my 42inch when playing games with my xbox controller for pc, i dont mind too much.
From what I can gather, I don't think HDMI is frame rate limited. The TV might be limiting your frame rate but that's the way that TV is set up, not HDMI in general. If connected to a PC monitor it should perform the same as DVI unless that monitor specifically has some factory settings or hardware limiting the frame rate of the HDMI port.
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