(i) Two elements of the 3d series for which the third ionization enthalpies are quite high are Zinc (Zn) and Copper (Cu). Both have completely filled d-orbitals, making it more difficult to remove the third electron.
(ii) Out of KMnO$_4$ and K$_2$MnO$_4$, KMnO$_4$ is paramagnetic. This is because in KMnO$_4$, manganese exists in the +7 oxidation state, which leaves it with unpaired electrons in the d-orbitals, making it paramagnetic. K$_2$MnO$_4$, on the other hand, has manganese in the +6 oxidation state, which does not result in unpaired electrons.
(iii) One consequence of lanthanoid contraction is the decrease in ionic radii as we move across the lanthanide series. This causes a similar size among elements, leading to similarities in chemical behavior.
(iv) Potassium manganate can be prepared from pyrolusite ore by reacting it with a suitable oxidizing agent like potassium hydroxide (KOH) under heating. The reaction is as follows:
$\text{MnO}_2 + 4\text{KOH} \longrightarrow \text{K}_2\text{MnO}_4 + 2\text{H}_2\text{O}$
(v) Oxygen stabilizes higher oxidation states of transition metals better than fluorine because oxygen has a smaller size and a higher electronegativity, which helps in accepting electrons from the transition metal. In contrast, fluorine, being more electronegative, tends to form more stable bonds in lower oxidation states.