My 05 does have the rough idle when I first start it in the morning, but the thing is I live in south florida so the cold weather is not the issue here. But every time I start my car in the morning it idles rough, together with engine movement and the CEL reading random/general misfire.
The rough idle is gone when I start it when the engine is already warm. The problem is with the cold engine. I can try doing this DIY cuz the parts are cheap, it is just labor intensive.
Oh and my car was supposedly protune by the previous owner.