Paternalistic Government policies
It looks right to me. There's an alternative (but equivalent) way to solve part D: we know that the consumer's choice will satisfy

$$ x^* =  \frac{ a(m-t)}{ p_x - s} ,\  y^*= \frac{(1-a)(m-t)}{p_y},  $$
or, with a balanced budget,
$$ x^* =  \frac{ a(m-sx^*)}{ p_x - s},\ y^* = \frac{(1-a)(m-sx^*)}{p_y}.$$

The government wants to maximize $\min\{x,y\}$, which is achieved when $x=y$. Solving
$$ x^*=y^*\iff \frac{ a(m-sx^*)}{ p_x - s}= \frac{(1-a)(m-sx^*)}{p_y}$$

for $s$ yields

$$ s^* = p_x - \frac{a p_y}{1-a}. $$