How would one go about finding the precise number of conjugacy classes for any given $n$ if $n$ is odd or even?
By looking at the cases of $D_6$, $D_8$ and $D_{10}$ I have some idea, but I'm not sure if it is right. Is there a procedure for figuring this out?
Consider a regular polygon with an even number of vertices, say a hexagon or an octagon. Each of these has two types of symmetry axis; one through pairs of opposite vertices, the other through pairs of midpoints of opposite edges. No symmetry of the polygon can move an axis of the first kind to one of the second.
Now consider a regular polygon with an odd number of vertices, say a pentagon or a heptagon. Each symmetry axis passes through a vertex and the midpoint of the opposite side. So one can rotate any of these axes into any other, using a rotational symmetry of the polygon.
These considerations should help you count the conjugacy classes of reflections in the dihedral groups.