Isn't (I^2)T always necessary to limit fault current heating, because heating is the danger?
Yes, so what's your point? You can ONLY use the MCP if there is also an OL relay, which is providing the I^2t protection.
To eliminate nuisance tripping the IT curve would have to follow the normal startup current curves but be shifted by some amount of I or T?
Not exactly sure what you mean here, but I'll try.
No, the
starting current must still fit within the same curve. The motor curve is a thermal damage curve, it knows not from whence the therms came

. But the
INRUSH current of the motor, that which represents the extremely short duration of magnetization current, can on occasion be too high for the magnetic trips of a circuit breaker and give nuisance tripping. Happens quite a bit actually, hence the need for adjustable instantaneous trips.
Tripping on I only would be for a very fast trip due to some huge fault current?
What do you mean by "I only"? It's all I (as in current).
But assuming you mean I as in Instantaneous, then yes, the Instantaneous tripping is there only to provide short circuit and ground fault protection, NOT overloading.
They are designed not to trip, ever, on ~115% of rated value?
Which "they" are you referring to? The instantaneous trips will not trip at even 800% of rated value. But assuming you switched to referring to the thermal trips of the TM circuit breaker, then it is 125% max and appears to be 115% minimum. But don't forget that you cannot load a breaker to more than 80% of its rating, because you cannot use wire that is sized for less than 125% of the load. (excepting 100% rated breakers for the purpose of this discussion). So although it appears that the breaker will allow 125% of rated current, continuously, it will not really be doing that in the field if you compare that to the wire ratings because you can only put a 16A load on a 20A MCCB. Remember, the breaker is there to protect the wire. The overload relay, which you will notice has a much lower threshold, is what is protecting the motor. In the case of a mag-only breaker + OL relay combo then, the amount of continuous current allowed to flow in the circuit is even lower.
The values are more-or-less insensitive to ambient temp as long as it's under some spec value?
All breaker trip values are shown at 40 degrees C and must be compensated for at temperatures above that. There is some argument as to the necessity to adjust for extremely low temperatures as well, I have yet to see a definitive answer on that. Altitude must be taken into consideration as well. Most are based on 2000m or less.