If we have the standard $2$-simplex (pictured below from Hatcher), why is there an arrow from $v_2$ to $v_0$? Why not from $v_0$ to $v_2$?
We have $\partial([v_0,v_2])=[v_2]-[v_0]$, so shouldn't there be an arrow from $v_0$ to $v_2$?
Also, why is the standard $2$-simplex oriented counter-clockwise?
First of all this all convention and definition. These are just the definitions that the author made here.
Pertaining to your questions: The idea generally is that boundaries of simplicies or other objects are "closed" in a certain sense. In your two-simplex we can picture this as following the path of the boundary and indicating the direction of the edges that are needed to end up at the original point.
Starting at $v_0$ you can go to $v_1$, then to $v_2$ and then back to $v_0$. Thus your path is $[v_0, v_1]$, then $[v_1, v_2]$ and then $[v_2,v_0]$. By convention $[a,b] = -[b,a]$. Then writing the boundary "algebraically":
$\partial [v_0, v_1, v_2] = [v_0,v_1] + [v_1,v_2] + [v_2, v_0] = [v_0,v_1] + [v_1,v_2] - [v_0,v_2]$.