Why did the train need to keep moving?
From a physical sense, the reason it needed to keep moving was that the train itself, as explained in the videos the children watch, was intended to be self-sufficient; from that, one must surmise that the train derives some unexplained benefit by being constantly in motion; perhaps this motion is necessary to somehow enable other, lesser, train functions, such as maintaining the conditions (channeling the heat from friction, perhaps?) for the interior of the train.
Note that, again from the videos, the train was not built with global freezing in mind; it was built as a luxury experience. Considering your example of the ice-breaking generating water, we can guess that if rain existed (as it does not in the film since everything is ice), the train would gather the rain in the same way it gathers ice, to replenish water supplies.
If the train didn't keep moving, whatever unexplained benefit it might be deriving from moving would be lost, and based on exterior conditions, the train might be frozen in place, rendered incapable of moving to re-derive any lost benefits. This dangerous and unpredictable scenario cannot be chanced, lest all of what is left of humanity be put at risk (from whatever environmental conditions might endanger the train; eg. an avalanche), so the train must keep moving.