Solve the equation \(( 1 + 2 \mathrm { i } ) w + \mathrm { i } w ^ { * } = 3 + 5 \mathrm { i }\). Give your answer in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(| z - 2 - 2 \mathrm { i } | \leqslant 1\) and \(\arg ( z - 4 \mathrm { i } ) \geqslant - \frac { 1 } { 4 } \pi\).
Find the least value of \(\operatorname { Im } z\) for points in this region, giving your answer in an exact form.