closure, convex hull and closed convex hull

No.

Let $A = \{(x,e^{-x})\}_{x\geq 0} \cup \{(x,-e^{-x})\}_{x\geq 0}$. Then $A$ is closed, and $\mathrm{co} A = (\{0\}\times [-1,1]) \cup ((0,\infty)\times (-1,1))$, which is not closed (take $(x_n,y_n) = (1, 1-\frac{1}{n})$).

Hence $\mathrm{co} A = \mathrm{co} \overline{A} $ is strictly contained in $\overline{\mathrm{co}} A = [0,\infty)\times [-1,1]$.

If $A$ is compact, the result is true (using, eg, Carathéodory's theorem).

Not necessarily. Let $$A=\Bigl\{(x,y) : y\geq {1\over 1+x^2}\Bigr\}$$ Then the closure of the convex hull is the closed upper half plane $\{(x,y) : y\geq 0\}$, but the convex hull of the closure is the open upper half plane $\{(x,y) : y > 0\}$.