When a man intends to start walking on the ground though the action and the reaction forces are equal in magnitude these forces may not produce accelerations of equal magnitudes. i.e the man undergoes acceleration but the ground does not get accelerated. Why?

Who says the ground (actually the earth) isn't accelerated? Every time you take a step you are pushing the earth and it moves accordingly. Of course the earth's mass is triilions of times more than you, so the magnitude of the resulting movement of the earth is that much less than the magnitude of your movement.

Yup.

Even the moon's gravity moves the earth as it rotates. Due to the distance and the size difference, it just makes little difference.