Multiple choice on sampling and aliasing
Question 1
Perhaps the easiest way to understand this is to draw $X(j\omega)$:

With the corresponding aliases after sampling in blue:

So long as the blue rectangle do no overlap you would not have aliasing. At this point it should be more clear that you need
$$
\omega_s \gt \omega_0 \\
T_s \lt \frac{2\pi}{\omega_0} \\  
$$
Question 2
Again, plotting $G(j\omega)$ (red), together with $75X(j\omega)$ and an alias image after sampling by $\omega_s$ (dashed blue) can help visualize the situation:

Indeed, because $\omega_{\max} &gt; \frac{1}{2}\omega_s$ you have some aliasing. However the question is not whether you can avoid aliasing, but rather to find the value $\omega_0$ below which the aliasing does not affect the spectrum.
In the above graph you may notice that the red and dashed blue curves overlap up to $50\pi$. So $\omega_0 = 50\pi$.