Is mirror a beam splitter with reflectence=1?

Now a mirror is just a BS with $r=1$. Therefore it seems that $U_{Mirror} = begin{pmatrix} costheta & 1 1 & costheta end{pmatrix}$.

This is incomplete. If $|r|=|sin(theta)|=1$, then $theta=pi/2$ and $cos(theta)=0$, so $$U_mathrm{mirror} = begin{pmatrix} 0 & 1 1 & 0 end{pmatrix}.$$

The Wikipedia page's assertion that the unitary corresponding to a mirror is a rotation matrix $R(theta)$ with variable $theta$ is incorrect.

That said, the details of what unitary needs to be used to describe a mirror will depend on exactly what labelling convention you use, so they cannot be given up front without that specification.