Okay, we can assume the death of the Emperor is cannon, regardless of what choice you made on the DB(destroy them or join them) as if you chose o detroy them there are still lose ends(Babette, Cicero and the Night Mother).

What effect do you think it will have on the Empire?

He was an old man so I would be surprised if he did not have an heir, but that is a sidenote.

I am surprised that the death of the Emperor has not affected the game more. Sure, you are in Skyrim and not in Cyrodiil, and they have plenty of their own problems (the death of the High King, the Stormcloak Rebellion,...) but the only faction that seem to care about Emperors fate are the Penitus Oculatus, and even they never stood a chance against the mighty Dragonborn. Imperials stationed in Solitude and around the Skyrim seem like they haven't acknowledged the Emperor's death (apart from some comments and notes).

In my opinion, his death is only cannon because the PC has no chance of seeing the impact of his acts on the Empire. Sure he must have heirs, even the Uriel Septim from Oblivion had them. Otherwise, the death of the Emperor would mean crisis in Cyrodiil, which would have a massive impact on Imperial units stationed in Skyrim.

I am only curious what we'll see in TESVI.

Even if he doesn't have heirs I'm sure someone else will be there to fill the void. Hopefully somebody worth it and not just one of the corrupt elder council members.

Think it's going to cause infighting again amongst the council and is the trigger for the thalmor's second invasion.

It may very well be canon, but it's not a forgone conclusion. Without the Listener, they Babette, Cicero and the Night Mother would have a rather small chance of ever getting the contract to kill the Emperor.

True, but the emperor's security would take a big hit after you "destroy" the dark brotherhood. It would make it easier for the dark brotherhood to regroup and strike.