Which unit - Discover Media, Discover Navigation, or Discover Navigation Pro.
I have the last unit (the 8" Pro), and whilst I have had a few random issues (freezing about 4-5 times since new), generally it's very well behaved, and I certainly have had no issues with the screen. Today it was only 3°C when I set off from work and the touch screen worked perfectly.
Out of interest, what is the current firmware on your unit (go to system information and under "Software" it will tell you your current firmware).
As regards the other start / stop warnings, I get these as well and think it's totally normal. Sometimes even when the AC is off, the system will pulse it to keep the refrigerant moving to lubricate the system and also if the Climate control detects a high amount of humidity, it can also override the AC switch and turn it on. Obviously when the AC compressor is running, the engine needs to be running.
Engine needs to be running is normally when the DPF needs a regen (either active or passive) and the Start/Stop system is overridden to keep the engine running to get the Cat up to temperature (I assume yours is a diesel) ? If so, then that's quite normal as well.
I've also found that having the temperature set to higher than about 23°C will cause the start / stop to be deactivated on the warm up phase (something to do with the PET electric heater), and lowering the temp to 23°C or below, usually kicks the SS system back in (on mine it does anyway).