Add a small invisible sprite just beneath the player's feet - if that is overlapping the ground, the player must be on the ground.
To check if the physics behavior has come to rest, check that both the X and Y velocities are close to zero. Mind out that it might not exactly equal zero - it might just have a very small velocity. Have some kind of threshold at which you consider the object at rest, like abs(X Velocity) < 0.01 or something.