The correct statement(s) about the redox and acid-base chemistry of manganese and chromium oxoanions is(are)

Manganate is stable only in strongly alkaline solution; in acid it disproportionates, $3\mathrm{MnO_4^{2-}}+4\mathrm{H^+}\to2\mathrm{MnO_4^-}+\mathrm{MnO_2}+2\mathrm{H_2O}$, so (A) is wrong. In neutral medium $\mathrm{MnO_4^-}+2\mathrm{H_2O}+3e^-\to\mathrm{MnO_2}+4\mathrm{OH^-}$ (3 electrons) while $\mathrm{I^-}+6\mathrm{OH^-}\to\mathrm{IO_3^-}+3\mathrm{H_2O}+6e^-$ (6 electrons); electron balance requires $2$ permanganate per iodide, so (B) is correct. Chromate and dichromate both contain Cr(+6); $2\mathrm{CrO_4^{2-}}+2\mathrm{H^+}\rightleftharpoons\mathrm{Cr_2O_7^{2-}}+\mathrm{H_2O}$ involves no change in oxidation state, so (C) is wrong. Mn in $\mathrm{MnO_4^{2-}}$ is $+6$, $d^1$, so it has one unpaired electron and is paramagnetic ($\mu\approx1.73$ BM); (D) is wrong. By contrast permanganate, Mn(+7) $d^0$, is diamagnetic, which is the likely source of the confusion. Only (B) is correct.