Why min AC = min SRAC at the minima of AC curve?
Short-run implies that some decision variable cannot be free set, it is fixed for a time (in the short-run). In the long-run all variables may be freely set. Let us denote the fixed variable by $x$. This can take different values, so there is a family of short run cost functions is $SC(x,q)$.
Given a value $q$ let us denote the cost minimizing value of $x$ by $x^*$, i.e.
$$
x^* = \arg\min_x SC(x,q). 
$$
Cost minimization implies that in the long-run this will be the chosen value of $x$, thus
$$
C(q) = SC(x^*,q) = \min_x SC(x,q).
$$
Dividing by $q$ we get
$$
AC(q) = SAC(x^*,q) = \min_x SAC(x,q).
$$
The above holds for any $q$.