Engine runs rough when cold

There are few possibilites, I am providing you with a list of them. Note that the distributor pick-up coil is a possibility.
I am also giving you a partial desription of ignition system.

The reason I am puttng together the symptoms and issues is that the tach gets its signal from the coil.
If, during start-up the timing is off or part of the Idle Air Circuit Control Valve, (IACV), is faulty, the timing can be thrown off which would cause the rough starting idle and the tach signal interference.
Timing is retarded some during warm up, if there are other issues in the start up, like the camshaft position snesor or IACV, the timing is thrown even further off than when warmed up.
I am providing you with a sdiagram of IACV circuit as well so you can see al parts invlolved insytem.

So, my thoughts are;
The coil is failing
The camshaft position sensor is failing.
The ignition timing is off due to worn out timing/beltchain.
The IACV is failing and causing a series of issues during start up. The IACV also works like a choke to let the air/fuel mixture lean out during start up.

However, the tach could be gettng a faulty signal from a lower than specified voltage signal form the rough idle which is probably lower than it should be. In other words, the information I am giving you is a good back up of possile failed vcomponents. The tach malfunction makes it a little harder to diagnose but makes me think it is an igniton control and/or timing issue causing the problem.
Also, the fuel pressure can cause rough idle as well as vacuum leaks.

The first thin to do is go to an Auto Zone or Avance auto parts, etc. And have them pull troubl ecodes. You do not have to have a malfunction light on to have a code. It will help in the process of eleimination of the source of the problem tremendously.

Let me know if you need any more help with this issue and I will reply ASAP.

Hello,

If it is a 4.3L V6 it could be the ignition coil it sends a tachometer reference directly to the tachometer from the primary side of the Ignition coil circuit it only make sense the coil is breaking down losing tachometer signal and weak spark causing the rough Idle. I'm attaching wiring diagrams and repair guides below applies to 4.3L v6 and 2.2L 4 cylinder the first 4 are for the V6 and the second 4 is for the 4 cylinder. The engine wasn't specified in the post. Now if we are working on a 2.2L 4 cylinder the tachometer receives the signal from the PCM. The PCM receives a signal from the Crankshaft Sensor. It could be 2 ground wires under the hood located left rear of the cylinder head make sure there not broke and clean and tight. The two grounds G112 and G114 in the wiring diagram. Let us know if this information was helpful and we are always here if you more. Note I spit the wiring diagrams in haft to enlarge for easier reading.

https://www.2carpros.com/articles/how-to-check-wiring

https://www.2carpros.com/articles/how-to-use-a-voltmeter

Thanks
JOe T.