Amino acids: why is charge found by comparing pH and pKa and not pH to pKa + log [A-]/log[H+]?
The properties you listed are correct, and all of them follow from the relationship described by the Henderson-Hasselbalch equation.
The equation shows a close relationship between $\mathrm{p}K_a$  and $\mathrm{p}H$. Note that $\ce{log ([A^{-}]/[HA])}$ equals zero when $\ce{[A^{-}]/[HA]}$ equals 1, and when $\ce{log ([A^{-}]/[HA])=0}$, $\mathrm{p}K_a=\mathrm{p}H$. This means that at the equivalence point, $\mathrm{p}H$ equals $\mathrm{p}K_a$.