Q: Extreme negative fuel trims only when coasting, with rpm bounce

Fuel trim is a very valuable diagnostic parameter but to evaluate fuel trim, you generally check at fixed engine speeds namely idle, 1,500 rpm and 2,500 rpm, for about 30 seconds. If the sum of STFT and LTFT is less than 10%, there is no reason for immediate alarm, although lots of variables come into play such as engine age and whether or not other malfunctions exist. The RPM fluctuation could be due to anything from a vacuum leak to a sticky throttle plate or malfunctioning idle air control valve. With regard to the rough idle when the engine is cold, on start-up the engine coolant temperature sensor must be working properly to signal the PCM to enrich the mixture on cold start. If there is a fault with that sensor, you can experience rough idle idle as you are describing. Other possibilities are uneven cylinder compression, a vacuum leak, an air induction fault like a stuck idle air control valve or a faulty throttle position sensor, problems with the mass air flow sensor, defective spark plugs, a weak coil and defective oxygen sensor(s). If you desire that the rough idle be diagnosed by a certified Mechanic, dispatched by YourMechanic right to your location, please request a rough idle diagnostic and the responding certified mechanic will get this resolved for you. If you have further questions or concerns, do not hesitate to re-contact YourMechanic as we are always here to help you.