1:
Ski troops basically only served in the german, russian, and finnish armies in any number. Of those, only the Finns would really make an appearance, as German and russian ski troops barely fought in combat.
2:
Since its Finnish, the ground in winter could be completly covered in snow. No grass to worry about being able to ski on or not.
3: Yes, it would be possible to have gathered inertia, much like the bicycle.
4: No, you can't fire while using skis. The armies did not train to fire while using skis. Even finnish ski troops used their skis only to get to combat, then dropped them, fought on foot, then rode off on the skis. Its too unwieldy, you can't get firm footing to fire a rifle while on skis, considering their point is to NOT have firm footing.