Numerical radius of a pair of operators in Hilbert spaces

For your first question, consider $$ C = D = \begin{pmatrix} 0 & 0 \\ 1 & 0\end{pmatrix}.$$ Then the LHS of the first inequality reads $$ \frac{\sqrt{2}}{4}\| 2\begin{pmatrix} 0 & 0 \\ 0 & 1\end{pmatrix}\| = \frac{\sqrt{2}}{2}. $$ For the RHS it is $$ w_e(C,D) = \sqrt{2} \sup_{\|x\|=1} |\langle \begin{pmatrix}0 \\ x_1 \end{pmatrix}, \begin{pmatrix} x_1\\x_2 \end{pmatrix}\rangle| = \sqrt{2} \sup_{\|x\|=1} |x_1x_2| = \frac{\sqrt{2}}{2}.$$ For your second question: Consider $$ C = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, D = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix},$$ then $Im(C) \perp Im(C^\ast)$ and $Im(D) \perp Im(D^\ast)$ but $$ \frac{\sqrt{2}}{4} \| CC^\ast + DD^\ast\|^{1/2} = \frac{\sqrt{2}}{4}, \quad w_e(C,D)) = \frac{1}{\sqrt{2}}.$$