Example of the semisimple ring $R$ but $R^{{\rm op}}$ is not. Is there any example of this kind of rings? i don't have any imagine of this rings, if they are exist!
By the Wedderburn-Artin theorem, a semisimple ring is a product of full matrix rings over division rings. Conversely, a product of simple artinian rings (like full matrix rings over division rings) is semisimple. This is left-right symmetric.
No. Every semisimple ring is isomorphic to its opposite ring, and this fact is a corollary of Artin-Wedderburn theorem. So if $R$ is semisimple, then $R^{op}$ is semisimple.