A wave is travelling along a string. At an instant, shape of the...

Since, $A$ is moving upwards, therefore, after an elemental time interval the wave will be as shown dotted in below figure. It means the wave is travelling leftward. Therefore, (a) is wrong- Displacement amplitude of the wave means maximum possible displacement of medium particles due to propagation of the wave, which is equal to the displacement at $B$ at the instant shown in figure. Hence, (b) is correct. From figure, it is clear that $C$ is moving downward at this instant. Hence, (c) is wrong.
The phase difference between two points will be equal to ($\frac{\pi}{2}$), if distance between them is equal to ($\frac{\lambda}{4}$). It may be equal to ($\frac{\lambda}{4}$). Hence, phase difference between these two points may be equal to ($\frac{\pi}{2}$).