(i) Zinc (Zn) and Copper (Cu) exhibit high third ionization enthalpies among 3d series elements. This is attributed to their completely filled d-orbitals, which require substantial energy to remove the third electron.
(ii) KMnO$_4$ is paramagnetic, unlike K$_2$MnO$_4$. In KMnO$_4$, manganese is in the +7 oxidation state, possessing unpaired d-electrons. Conversely, manganese in K$_2$MnO$_4$ is in the +6 oxidation state, which lacks unpaired d-electrons.
(iii) Lanthanoid contraction results in a progressive decrease in ionic radii across the lanthanide series. This leads to elements having similar sizes and consequently, comparable chemical properties.
(iv) Potassium manganate (K$_2$MnO$_4$) can be synthesized from pyrolusite ore (MnO$_2$) by reacting it with potassium hydroxide (KOH) under heating conditions, as shown by the reaction:
$\text{MnO}_2 + 4\text{KOH} \longrightarrow \text{K}_2\text{MnO}_4 + 2\text{H}_2\text{O}$
(v) Oxygen is a more effective stabilizer of higher oxidation states in transition metals compared to fluorine. This is due to oxygen's smaller atomic size and higher electronegativity, facilitating electron acceptance from the transition metal. Fluorine, while more electronegative, tends to form more stable bonds in lower oxidation states.