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

(img src: http://jan.ucc.nau.edu/~doetqp-p/courses/env440/env440_2/lectures/lec19/lec19.html)