'Hard' soap vs. 'Soft' soap: Why do they work this way?

In school, we did an experiment using fat and $\ce{KOH}$ to create 'soft' soap.

Afterwards, we added $\ce{NaOH}$ to change it into 'hard' soap.

I read on multiple places on the internet that the difference between using Potassium and Sodium is what determines if it becomes a 'hard' soap or 'soft' soap.

Am I correct in the notion that the $\ce{K+}$, resp. $\ce{Na+}$ ions connect to the $\ce{O-}$ ends of the Fatty Acid?

Is it possible to skip the first step and create 'hard' soap by using $\ce{NaOH}$ right away?

Why does 'Soft' soap dissolve so much easier than 'Hard' soap? Or in other words: Why is it much easier for the $\ce{K+}$ to fall off than for the $\ce{Na+}$ ?

Basically, yes, by common terms, soft soap has $\ce{K+}$ and hard soap has $\ce{Na+}$.

They are both alkali metals, but $\ce{K+}$ is below $\ce{Na+}$ on the periodic table so maybe we should consider periodicity trends. As you go down a table the atoms become more reactive, so $\ce{K+}$ is more likely to react with polar parts of water than $\ce{Na+}$. Here are solubilities for both (though there are other values that would tell this story, perhaps better than solubility of the product):

[Solubility for $\ce{KOH}$] in $\ce{H2O}$ at 25C: 121 g / 100 mL

[Solubility for $\ce{NaOH}$] in $\ce{H2O}$ at 25C: 100 g / 100 mL

Regarding your question of going straight into hard soap, based on this link it seems doable; he just used $\ce{NaOH}$ (lye) and oils with no mention of $\ce{KOH}$ (potash?) to speak of.

And yes, the positive ions connect to the $\ce{O-}$ of the fatty acid as seen in this diagram. • This is not an answer to the question, and the part about the reactivity is also complete nonsense here. – Karl Aug 31 '17 at 21:28