(a) In ${\left[{\mathrm{NiCl}}_4\right]}^{2-},\mathrm{Ni}$ is in +2 oxidation state and each ${\mathrm{Cl}}^{-}$donates a pair of electron. So, ${\mathrm{Cl}}^{-}$acts as a weak ligand and does, not cause any forced pairing. Thus, electrons remain unpaired making it paramagnetic.

In $\left[\mathrm{Ni}{\left(\mathrm{CO}\right)}_4\right],\mathrm{Ni}$ is in zero oxidation state and $\mathrm{CO}$ acts as a strong ligand causing forced pairing. Thus, no electron remains unpaired making it diamagnetic.
(b) (i) ${\mathit{t}}_{2\mathit{g}}{}^3{\mathit{e}}_{\mathit{g}}^2$
(ii) ${\mathit{t}}_{2\mathit{g}}{}^5{\mathit{e}}_{\mathit{g}}{}^0$