I play on a 32" TV @ 1920x1080 over HDMI and what I found with COH2 was there were times when the Videocard/Game identified the TV as a TV and dropped the refresh rate to 24 Hz. And it was amazing how laggy it was. I could tell something to move and it wouldnt move until 2-3 seconds after I told it. The TV would be 60Hz in Windows, then switch to 24Hz in game...puke.
You can force the refresh rate in your Steam startup. Under the steam games properties you will have:
SET LAUNCH OPTIONS
add the line -refresh 60
just -refresh 60 ?