Gas Pedal Takes Time To Respond, Focus 2010 Mk2.5

Hi there,

I have a Focus 2010 Aut, MK2.5 Europe version.
The problem is in the pretty cold mornings, when the temperature is around 50F-60F. When I start the engine, the RPM goes up, then falls over 600 and the engine rumbles like it was dying. Sometimes it recovers and very few times it dies, so I need to start again.
No matter if the engine recovers or I need to start it again, the very first seconds I drive, when I press the gas pedal it takes time to respond. If I press the pedal only around 50%, the engine rumbles a little, like it is going to die. Not at the same level as I explained above, but it rumbles a little. The RPM goes to 800 and keeps there until I press the pedal at full.
Also, if I press the pedal at full, it takes time to respond, around 1.5 seconds.

All of these "issues" last for maybe 3 or 4 events/gas pedal pushes, then it goes to normal condition. No more rumbles or pedal delays.

I already changed spark plugs, cleaned fuel injectors, checked wires, transmission level fluid. The condition remains in cold mornings. When the weather is warm, the car is great, no problems at all.

It is pretty hard to find information for this car. Hopefully you can help me with this issue.

Thank you.