Finite groups as subgroups of dihedral groups

No, because every dihedral group $D_n$ is solvable, and hence every subgroup of $D_n$ is solvable, too.

Actually, more can be said. Every subgroup of $D_n$ is either cyclic or dihedral, see Theorem $3.1$ of K. Conrad's notes. Therefore, say, $Q_8$ cannot be embedded (which we see by Doeke's answer).

No. Dietrich Burde's reason is quite nice, but there are easy elementary concrete counterexamples. Here's one:

$Q_8 = \{\pm 1, \pm i, \pm j, \pm k\}$ has six elements of order $4$, but no dihedral group has this property: all reflections in a dihedral group have order 2, and the only rotations of order $4$ are rotations by $\frac{\pi}{2}$ and $\frac{3\pi}{2}$ so a dihedral group has at most two elements of order $4$.