I will try to answer every question, but feel free to ask if anything still confusing, there are no dumb questions!
About the Resolutions System
The HD is like an abstraction of the kid's will to play. Think of it as your health already, and you're always rolling against yourself (just like real life!).
So, lets take the dwarf as an example, their HD starts with 1d12. Every time you roll for anything you roll a 1d12 +something. If the result was below 10 but above 8, your HD till end of the session, or till you do a long rest will be a d10. If, for example, the roll was below 6, but above 4, aka 5, your HD will be for now on 1d6.
Some stuff, even narrative inputs by the DM, can add a negative modifier. Making it easier for your HD to shrink to a die with fewer sides. Once you hit the d4, if you roll below 3 that dice is lost. Once you level up you get one more HDs for your character, like extra lives.
About Combat
So, that's kinda an extension of what I said above. You don't roll to attack, just to defend. Same happens to monsters, the defender rolls their HD +Armor and subtracts the weapon value of the attacker. Stronger weapons make it harder to defend yourself.
By this point, if I made myself clear enough, you maybe were thinking: "So it's kinda hard to kill someone in combat?"
You're right, because they are kids larping, they are not hitting "for real". But doing other stuff can get you bored, so the rule of the thumb is to roll when there is risk or need of impartiality.
About Saves
Saves are kinda a "free off jail card", when you use a save you don't roll! Simply get the best possible outcome.
I hope that was helpful!