The car is a 2002 Audi A4 1.8t Quattro.

Problem: when climate temperature goes below 0 celsius, car has rough time starting.

I will attach an mp3 file where you can hear the issue, but pretty much when starting the engine it cranks then stops, then cranks and stops and continues until it barely starts. Like the starter is lagging or something.
It always fires up eventually, but recently it feels as it's gotten a lot worse.

Only happens when cold starting.

Already replaced the starter, replaced coolant temperature sensor that's in the back of the engine as I got a code:

17704 - Error in Mapped Cooling System (check Temp-Sensor and Thermostat) P1296 - 008 - Intermittent

^Should I have replaced the coolant temperature sensor under the radiator istead?

Also sometimes I get CEL for car running lean (P1136), but that comes on maybe once every 1-2 months.

I have access to VCDS, just not that experienced with it.. I believe I could check some crucial values there but don't know how.

Any help or info if someone has experienced same or similar issues is very appreciated.