A Parabola with a Median Chord

The parabola $C:y^2=2px$ ($p>0$) has focus $F$ and directrix $l:x=-1$. Points $A(x_1,y_1)$ and $B(x_2,y_2)$ (with $y_2>y_1\geqslant 0$) lie on $C$, the line $AB$ meets $l$ at a point $M$, and $A$ is the midpoint of $BM$.

1. Find the equation of $C$. 2. Find the minimum area of $\triangle BFM$. 3. If $\angle MFB=\dfrac{2\pi}3$, find the coordinates of $A$.