Can the Moon provide momentum to an object in Earth's orbit? Gravity Assisted Boost Can a satellite maintain an equatorial orbit around Earth near the Moon's orbit to receive partial gravitational boosts by gaining momentum as the satellite passes though the Moon's gravity well? 

I'm not asking about hanging out in the Lagrange points, but passing between the L2 point and the Moon or the Moon and Earth aligned, in a Moon assisted orbit.
The Moon has a "Roche lobe", where the Moon's gravity dominates, and all you need for the situation you describe is that the Roche lobe lies physically outside the Moon.  That will be true, because the Lagrange points are on the outside edge of that Roche lobe, and it's outside the Moon.  The Moon's Roche lobe is shown here: http://hyperphysics.phy-astr.gsu.edu/hbase/Mechanics/lagpt.html