Protonation of Guanidine
Your misconception is directly at the beginning of your question and turns up multiple times. There are no $\mathbf{sp^3}$ nitrogens in guanidine.

Guanidine is isolobal to urea, where the carbonyl oxygen has been replaced by an imine $\ce{NH}$. However, in principle it is still the same flat, resonance-stabilised molecule. The main difference is that there is no ‘preferred’ site for the double bond — it could point towards any of the three nitrogens in theory; you could say the diamide resonance is enhanced.

This is also why the $\ce{-NH2}$ nitrogens are not basic. Their lone pair is not sitting in an orbital all alone waiting for a proton. It is actively resonating with the &#x3c0; system of the other two. The only available lone pair guanidine has is that of the $\ce{=NH}$ nitrogen sitting in an $\mathrm{sp^2}$ hybrid orbital. Thus, a second protonation requires somewhat superacidic media.