Given that our only glimpse of dwarven society is the life of one disabled veteran and her young son, I think it's a bit of an leap to assume that there are no wars. Or at least, no violent conflicts: Sigdi's husband and a lot of innocent miners died in a troll attack, after all, with no war required. Monsters are everywhere, especially in underground tunnels.
While the line about picking a fight with a tree was intended for comedy, the point still remains that if you know for certain you will only get a good afterlife if you die with honor, you are far more likely to deliberately go seek out an enemy to fight—whether it's orcs, giants, drow, duergar, etc.—as soon as you know you're sick or getting old. I imagine honor duels are probably a very common way of settling the score between two feuding dwarves: If I win, then I win, but if I lose, I still get an honorable death. And again, if you're sick, why not provoke a duel you know you will lose?
Also, there are certainly ways to die with honor that don't involve combat. If there's a cave-in and you take the time to save another dwarf but can't get out yourself before it collapses, that's an honorable death even though you were killed by a falling rock.
Basically, this all comes down to the fact that the dwarves know, for a fact, that this is how the world works for them. There's no reason to assume that they don't have methods for dealing with it.