Did Kurama become a whole after the last war? Did Kurama get his half from Minato and become a whole after the war? Thus, in Boruto, is Kurama composed of yin and yang already?
Yes.

During the war, Black Zetsu absorbs Minato's half of Kurama -- Minato tries to revive Naruto (from whom the other half was just extracted by Madara), but Zetsu steals it, and that is how the Gedo Statue becomes complete. Here is that particular scene on YouTube.

After the war, the "whole" Kurama goes back inside Naruto.