Basically, yes, by common terms, soft soap has K+ and hard soap has Na+.
They are both alkali metals, but K+ is below Na+ on the periodic table so maybe we should consider periodicity trends. As you go down a table the atoms become more reactive, so K+ is more likely to react with polar parts of water than 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 KOH in H2O at 25C: 121 g / 100 mL
Solubility for NaOH in 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 NaOH (lye) and oils with no mention of KOH (potash?) to speak of.
And yes, the positive ions connect to the 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)