They never showed Chiefs face. They show his eyes..... so what? That doesn't show what he looks like. The only thing it tells us is that he's white. That's it.
If you listen to Cortana before the Didact fires the Composer, she tells us that there is a slipspace portal opening. This is what the Didact falls into. It is entirely plausible and highly probable that he survived.
As for Cortana, she saved John by manipulating the hard light bridge to create a shield around him. There are hidden hints in the campaign suggesting that Cortanas rampancy can be beaten and the "accepted" 7 year life span may not apply. This is from a recording in Dr. Halsey's log that can be found in the wreckage from the Forward Unto Dawn after you crash on Requiem.
The Didacts narration in the epilogue is not after the event of Halo 4, it is long before. It is him addressing the forerunner council after using the composer to enslave humanity as his army to defeat the flood. He was stopped by the librarian before he could convert them all into the Prometheans. The monologue at the end would be his final words before he was sentenced to the Cryptum and locked inside Requiem with his own army used to keep anyone from releasing him. After that the Halo rings were fired and we all know what happens next.
By the way, I have not read the books yet. Most of this information is somewhere in the game. Between the 7 terminals, hidden computer logs, multiple campaign completions, and just a little wiki, the grand picture comes together. I loved every moment of this game and I have been playing since the first one came out. It is obvious that there is a bigger picture that 343 is working towards and I have 100% faith in them after playing this game.
My suggestion is to go back through the campaign. Take your time to explore the levels. Look for hidden information (I have found 3 just in the wreckage of the FUD). Watch the cut scenes very closely. Listen to the in game conversations. There is way more than you expect, and it is very rewarding to discover the wealth of information that is scattered around.