The low pressure warning uses differential rotation times so in hard cornering you could probably get it to think they were rotating at different speeds not because you are cornering, but because they are flat. That's one of the limitations of an ABS-based pressure monitor.
The ASC+T light goes off all the time on my car but I've never noticed the flat tire light. That said, my eyes are usually on the road and the tach (where the traction warning light lives) for corners, not the speedometer where the flat indicator is, at least for Canadian Coopers.
The ASC+T light is supposed to be a "you're going too fast, slow down" indicator but I usually take that to mean I'm going just fast enough. If I take 'em fast enough that it actually has to kill the engine and/or the tires actually slide, then I know I'm going too fast... In spirited driving on dry pavement I find it is smoother (no brakes applied or engine retarding) to turn it off but if there is ANY water on the road at all then I leave it on as it is very easy to go around a blind corner and find a big puddle. So far I'm quite impressed with how well the ASC can haul in a wet corner. I should probably back off my driving some but it is so much fun and the car feels so glued to the roads that it is hard to take the "ASC+T can't break the laws of physics" advice!
Back to the posted topic, engine braking does work but with those nice big brakes, I like to heel and toe to use both engine and wheel brakes. After 4500km the brake dust has finally started to slow down on my car. Now I can tell too when I've braked hard; there will be an obvious irregular deposit while normal braking just leaves a light dusting
Harry