What is the point of monsters having damage modifiers that are higher than their attack die? For instance 1d3-4. Does that mean that the creature only does damage on a critical hit if its die roll equals more than 4? Is there a rule that nullifies the negative if the creature attacks another creature smaller than it?
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If damage would be reduced to less than 1, then the creature instead deals 1 point of nonlethal damage.
Creatures are still statted because they may become the target of size increases or other buffs, and we need to be able to calculate what their new damage becomes.
Say that 1d3-4 was your Familiar. You use Share Spells to cast Enlarge Person and Bull's Strength on it. Its damage would increase to 1d4-1. Still not great, but if it had not been listed at all then we wouldn't know where to begin.
What is the point of monsters having damage modifiers that are higher than their attack die? For instance 1d3-4. Does that mean that the creature only does damage on a critical hit if its die roll equals more than 4?
No, when you have a critical hit, you multiply the modifiers, too. If an ogre (2d8+7) confirms a critical hit, he does 4d8+14 damage. Likewise, if a tiny creature with 1d3-4 damage confirms a critical hit, it will roll 2d3-8 damage (and it will still do just one point of non-lethal damage, though I like to apply the critical multiplier to that non-lethal damage so they get some benefit from confirming a critical hit - but I think the RAW is calculate the critical damage fully, then if it's less than 1 HP, do 1 non-lethal damage).
Is there a rule that nullifies the negative if the creature attacks another creature smaller than it?
No such rule. Theoretically, that smaller creature has really small HD and low HP already. Those low HP are how the game represents that it's easy to kill. It doesn't need extra rules to make it even easier to kill, and no such rule exists.