Current waveform in an inductor with AC source
Actually, the current is lagging the voltage by 90 degrees throughout the entire graph. What is confusing you is that the current has a DC offset. This DC offset is actually the 'transient' solution, and it does not die off because there is no resistance (dissipation) in this circuit.


  Second confusion is that for the first half cycle of the voltage in graph "A", current is rising both for increasing and decreasing voltage. How is this possible that current is rising in the inductor for both increasing and decreasing voltage ?


This is the behavior you expect because of the relationship between voltage and current:

$$V = L\frac{dI}{dt}$$

So if V is positive, \$dI/dt\$ is positive, and therefore the current I is rising. I is increasing any time V is positive, not when \$dV/dt\$ is positive.