With 120HZ you should see nothing but crisp clear high fps. That's what I see on my 52 inch 120 hz.
Common misconception...that has nothing to do with the 120Hz refresh capability. Most input sources are 60Hz sources unless they're outputting for 3-D glasses. When watching non-3-D TV, DVD, or Blu-Ray the only way to utilize that 120Hz refresh rate is by enabling frame interpolation post-processing in the TV (which IMO is only good for sports...TV shows and movies look like crap with frame interpolation turned on). Most console games are frame-locked to 30fps, so the 60Hz limitation should be having absolutely no effect on framerates.
Panel response time actually can make a difference in visual quality when things are moving quickly in the video, but that has nothing to do with the refresh rate. Having the console framerate drop below 30fps can also cause visual problems (frame tearing) because it breaks the vertical sync with the display, but that's an issue with the game/console itself rather than the TV.
Both HDMI ports are the same. Do you think a different brand would be better? Seems it would be a big hassle to exchange though.
If it's the fault of the LCD response time then your only recourse would be to return the TV, unfortunately. In the settings for the TV is there a "Game Mode" setting? This could be a global setting or it could be on a per-input basis. If not you could check to see if "TruMotion" (LG's name for frame interpolation post-processing) is enabled and see if you can try turning it off.