= Solution
The <conjugacy class> of $g$ is $\{hgh^{-1}:h\in G\}$. Rotations commute with one another, while conjugation by a reflection changes $R^j$ to $R^{-j}$. Thus the rotation classes are $\{I\}$, $\{R,R^4\}$ and $\{R^2,R^3\}$.
For reflections,
$$
R^a(R^jm)R^{-a}=R^{j+2a}m.
$$
Because multiplication by two permutes the residues modulo five, these conjugates exhaust all five reflections. Consequently the complete class partition is
$$
\boxed{\{I\},\qquad\{R,R^4\},\qquad\{R^2,R^3\},\qquad\{m,Rm,R^2m,R^3m,R^4m\}.}
$$
Their sizes $1+2+2+5=10$ account for every element, as in the <conjugacy classes of a finite dihedral group>.
A <normal subgroup> is a union of complete conjugacy classes. The trivial subgroup is normal, and the rotation subgroup is normal because conjugation preserves it; equivalently it has index two. None of the five order-two subgroups is normal, because it contains just one member of the five-element reflection class. Thus the proper normal subgroups are exactly
$$
\boxed{\{I\}\quad\text{and}\quad\langle R\rangle.}
$$
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