Is this when in idle?
Perhaps you need to replace your spark plugs, maybe if it is fuel injected, you may need to clean the fuel system.
sounds like a vacuum leak
Have vehicle scanned to determine the problem
I have 1998 a Lincoln town car with the same problem.. It idles rough when you start, then smooths out once it warms up and you give it gas. From what I am reading its either an oxygen sensor or you have vacume leak in one of your hoses by the manifold.
Could be a crack in one of the power pack tops , spark plug wire or wires. Just a couple of things that you didn't check
1 of the bolts has come out of your pressure plate or flex plate. i have had this prob on several vehicals over the years
Seeing as it is a diesel engine it has to warm up first in the colder weather. You say that you are starting the engine early in the morning? Well the black smoke may be caused by he motor being cold or simply by the fact that it is a big diesel engine. The rough ride you are referring to is caused by the motor being under-heated, and then as the engine heats up it smooths out.
Runs rough, water coming out of tailpipe if I was a betting man I would bet my last twinki that you have a cracked head or head gasket. water is crossing over into your exaust valves causing the water to come out of your tailpipe and the decreased power in the runs rough is from the compression leak. I am willing to bet that when your engine is running and your radiator is full along with your overflow you have bubbles blowing into your overflow if everything is not full fill it annd check.
it runs a little rough at first!
try checking out the coil pack
If the engine runs rough after a timing belt change, the belt may not have been tightened properly. It may also mean that the wrong timing belt was installed. Either way, it needs to be taken back to the mechanic.
Your timing is off. Buy a timing light.
because that is the way it runs