А composition of rotations is ether rotation or translation (a special case of Chasles' theorem) depending on the total rotation angle which is $m\pi /n.$ (This rotation angle can be controled by complex numbers as in Philipp Lampe's comment). So it must be a rotation for $m\not\equiv 0\pmod {2n}.$ If $m\equiv 0\pmod{ 2n}$ it must be a translation (but translation vector may be zero).

А composition of rotations is ether rotation or translation (a special case of Chasles' theorem) depending on the total rotation angle which is $k\pi /n$, so $P_k=R_0^{k\pi /n}(O_k)$ (rotation with some center $O_k$) for $k\not\equiv 0\pmod {2n}$ and $P_k=T_{v_k}(P)$ (translation) for $k\equiv 0\pmod{ 2n}$ (This rotation angle can be controled by complex numbers as in Philipp Lampe's comment). So fixed point exists only if $k\not\equiv 0\pmod {2n}$ (this point is $O_k$) or $k\equiv 0\pmod{ 2n}$ and $v_k=0$ (all points are fixed). The last case can be checked starting from arbitrary $P$ and checking whether $P_k=P.$