Effects of change in tax on output
you can do this using comparative statics and envelope theorem. The profit function specification was correct given the assumptions, let me just rewrite it as:
$$\pi (q,t) = p q - C(q) - tq$$
However, given the above we also know that the optimal quantity will depend also on a tax rate, so we will have $q^* = q(t)$.
Now by envelope theorem:
$$\frac{\pi^*(t)}{dt} = \pi'_2(q^*(t),t) = - Q^*(t) $$
So the optimal level of output given the tax $t$ will depend inversely  on change in profit when the tax rate changes or  $-\pi'_2(q^*(t),t) = Q^*(t)$.