Why/how does resonance stabilize a molecule?
One common misconception is that molecules interchange between resonance "states." This is not true. A molecule exists in
all resonance states
simultaneously. The whole idea of resonance is that we invoke it because we cannot, using our quite primitive way of expressing molecules as sticks and letters, express truly how some molecules exist. So we need a better representation and resonance was devised as that better representation, with the caveat that it is
only a way of thinking about the molecule and the molecule doesn't actually exist in any of those states in isolation.
So then it becomes clear why resonance stabilizes a molecule. Imagine trying to walk around on your tippy toes versus walking around on your feet. Which one is more "stable" for you? When a molecule resonates between various forms, it is literally spread between those states and this spreading of energy results in stability (like standing on your foot). When a molecule has no resonance forms, it is stuck in one form and thus is comparatively less stable than its resonating analog (like standing on your tip toes).
Why do delocalized electrons make a molecule more stable than one with localized electrons?
This is related to the above. Delocalized electrons are spread spatially and thus energy is dispersed spatially, which stabilizes the system. In many cases, that energy can be delocalized over many atoms, some of which might "want" the electrons more than others - that stabilizes the system. So imagine that in a localized system, the electrons would be on carbon which is not that electronegative. But if those electrons can be pi-delocalized to a nearby oxygen, that would stabilize the system because oxygen is more electronegative than carbon.
Why is acetic acid a stronger acid than phenol? (other than having a smaller pka, what about the structure makes it a stronger acid in terms of resonance and electrons?)
Your question here has less to do with resonance than by the mere fact that acetic acid is carboxylic acid and phenol is an alcohol. There is a severe electron-withdrawing effect of the carbonyl that renders the hydroxyl proton much more acidic. It doesn't have to do with resonance much. A better question, if you wanted to isolate resonance, is why phenol is much more acidic than other alcohols.
Acidity is measured by how stable the conjugate base is. So your question becomes, why is phenolate more stable than other alcohol conjugate bases? I think you know the answer to this question. It's because the phenolate's extra lone-pair can be delocalized into the phenyl ring, which stabilizes the system as noted above. In aliphatic alcohols, for example, there is no resonance and so the concentration of charge is less stabilizing (think of standing on your tip toes here). That's why phenol has a pKa of 10 whereas most alcohols have pKas in the 17 range.
