Hello, since vehicle seems to run better cold, overall miss, and better when o unplugged I can say the engine is most likey running rich. This being due to the fact it runs better cold when "rich" is required. Now you need to determine why running rich. have you tapped on the MAF ( mass airflow sensor) gently? this is the large in hose sensor between air cleaner and throttle body. Also we will need to check fuel pressure and also see if pressuree holds when key is off. Another test would be to lightly spray around intake, throttle body , egr with carn cleaner. If idle smooths while spraying, that would be an area of a vacuum leak. Vac leaks cause the computer to make engine run richer..
Here are common reasons for running rich.MAF sensor
Vacuum leak
Injector leak
bad coolant sensor
High fuel pressure
If you hook up a fuel pressure guage, crank engine, turn key off..... pressure should hold for good while. If it drops quick, most likely a leaking fuel injector. A coolant sensor can be tested with a scanner to see true degree temp. Tap on MAF and if engine runs rougher or stalls, need new sensor.
Hope this helps as a guide. LMk Thanks
ASE Certified Technician
32 years Foreign / Domestic Cars All aspects of Diag and Repair