It is normal for the ICE to start up on normal acceleration. You want the ICE to start unless you will just be costing to the next stop or cruising a parking lot; otherwise more fuel will be used to recharge the battery than would have been used to accelerate.

Yes, if the SOC is high, the TCH will allow more acceleration in EV mode. If the SOC is too low (under three bars), any acceleration will start the ICE.

I find I get better mpg if I accelerate normally (using the ICE) and then maintain speed with EV mode, than taking a half mile or so to come up to speed using only EV mode.

Tip: After the car is warmed up - allow the car to stop with the ICE idling (if necessary, press the gas pedal while holding the brake pedal down to start the ICE) and let it "shut off". It will take 8 to 10 seconds. After the ICE shuts off at a stop, the hybrid system switches to full mode and freely switches to EV mode anytime you coast at any speed under 40mph. I find it easier to "start" the ICE while coming to a stop by shifting to "B" around 25mph. Don't forget to shift back to "D" after stopping or the ICE will not shut off.