How does the summoning of the Ten Tails work? If Demonic Statue of the Outer Path requires only a part of the chakra of the Nine-tails and the Eight tails, why did they have to kill the other Jinchuuriki to obtain the tailed beast just to complete the Juubi? 

Why not obtain only parts of the chakra of each tailed beasts?
Because the Juubi will not be revive in its full form and make the juubi weaker for every tailed beasts that has been extracted only by parts.