I have been told that only those who inherit the highest degree of the Celestial Kingdom will be able to have continued posterity. Will those that don’t attain this level be allowed to procreate and have continued posterity? Or will the powers of procreation be taken from them?
Only those who are sealed as husband and wife in the temple, and who live by their covenants, so that they will be exalted in the celestial kingdom, will continue to procreate. The children that they will bear in that exalted state will be spirits, and will need to go through the mortal experience as we have done in order to gain physical bodies. The answer to your question is contained in D&C 132:15-17:
15 Therefore, if a man marry him a wife in the world, and he marry her not by me nor by my word, and he covenant with her so long as he is in the world and she with him, their covenant and marriage are not of force when they are dead, and when they are out of the world; therefore, they are not bound by any law when they are out of the world.
16 Therefore, when they are out of the world they neither marry nor are given in marriage; but are appointed angels in heaven, which angels are ministering servants, to minister for those who are worthy of a far more, and an exceeding, and an eternal weight of glory.
17 For these angels did not abide my law; therefore, they cannot be enlarged, but remain separately and singly, without exaltation, in their saved condition, to all eternity; and from henceforth are not gods, but are angels of God forever and ever.