Sorry to hear your misery, but hey Bimmer are so sensitive, if they felt a hint of betrayal they will let you know
Anyways let me shed some lights into your trouble.
The coolant warning pops up randomly, do you mean only at start up in the morning? If so the coolant maybe a bit low. On a cool engine (don't do it when engine is warm, you may hurt yourself), open the reservoir and there should be a red stick sticking out, if not your coolant is low and need a top up. I used to top up for my E60 every few months despite there's no water on my parking spot. The car not only consume oil, it also consume water probably because of the high compression rate they run at. Just distilled water is fine you don't need to mix coolant this and that and don't put mineral water in it
For the tire warning, it will only prompt if there's a slow leak, because our car doesn't have the TPMS in the wheels like the Americans do, it only senses the wheel spin relative to the other, not going into the tech stuff but conclusion is, it only senses graduate pressure lost not sudden lost.
In your case sounds like a sudden lost of pressure.
If you are going into service then I am sure BMW will take care of everything for you.
Good luck then.... oh did I mention the CLS looks hideous? The rear light cluster looks like two squash pea with cheap LEDs. If I have offend any Merc owner then... good