I think that the Ur-Didact did not die. There is proof of this as well. He was alive when the grenade was detonated in his chest. Then he fell into a slipspace rupture, which wouldn't kill him. It would just teleport him somewhere else in the galaxy. As for Cortana the chances of her surviving are slim. Although a very dark theory I have that could be why Halo 5 is exspose to be so dark. I think the Didact retrieved part of Cortana when he came into contact with her. I think the Didact, along with Cortana was teleported to a Forerunner Installation such as one of the Halo rings. I think Didact modified and fixed Cortana and forced her to serve him not humanity, not even her close friend. Then she would betray her former masters, humanity, and serve the Didact instead. This is just my opinion though.
\Also I know the Didact hates Humanity but maybe he would activate the Halo Rings to kill Humanity. I know he saw the Halo rings as a sin but maybe he just looses sight of everything in Halo 5 and kills humanity at any cost. The Didact taking Cortana can be supported by the fact that he said in Halo 4 I sense your malfunctioning companion human, and yet, she eludes me. This clearly shows intrest towards the AI. Also in the final cutscene the Didact states to Cortana "Your compassion for mankind is misplaced." This shows that he did not see her as an enemy, but a useful resoursce if on his side. It is also further backed up, that he said the line trying to persuade the AI to help him, not humanity.Altough this is still just my opinion. Although I do think that many agree, the Didact is not dead, and will be in Halo 5. Leave a comment below stating what you think.