3) Kuzan is aiming for Raftel:
I believe destroying Blackbeard or the WG is not Kuzan's objective. He is simply trying to reach Raftel by allying himself with Blackbeard. I will elaborate my thoughts here.
It is a crime to conduct any research on void century and deciphering poneglyphs. Due to the propaganda of the WG, the marines believe that anybody studying void century is basically trying to revive the ancient weapons and destroy the world. It is clear from what Nico Olivia's statement that the marines are a subject to this indoctrination =>
Kuzan was also a firm believer of this. He was a man who followed "burning justice" and believed in laws of the WG. He was willing to be ruthless for the greater good =>
However, things changed since the Ohara incident. Saul was a friend of Kuzan by his own admission, but Kuzan had to kill him at Ohara. He also despised the fact that a ship of civilians was massacred by Akainu. I believe Kuzan could never get over the horrors of Ohara and couldnt justify his conscience of killing innocent civilians. If you think about it, Kuzan's hatred towards Akainu's action at Ohara eventually culminated in the unprecedented duel of the admirals. Kuzan was initially indifferent to the fleet admiral appointment, but he fought a death battle just to stop Akainu from shaping the marines under his extreme vision. Kuzan saw with his own eyes how (I am quoting him) "
...complete and unyielding justice... can turn a man to madness sometimes".
The death of Saul and the massacre at Ohara impacted Kuzan to the point he let Robin escape and closely followed her growth afterwards. I believe Kuzan was trying to keep a small Ohara alive in Robin. He told her, "
...then keep yourself alive...And Ohara... hasn't fallen."
The point I am trying to make is Kuzan's change to lazy justice, his duel with Akainu and eventual resignation from the marines all lead back to Ohara. Thats why it wont be far fetched to believe Kuzan is highly interested in knowing the real reason behind the Ohara massacre, i.e., the void century and ancient weapons.
In chapter 967, it was explicitly mentioned that the Roger crew learned (a) the 100 year history of the void, (b) what the people of D, (c) what the ancient weapons are, and (4) the treasure left by Joyboy once they reached Raftel.
Thats where Blackbeard comes into Kuzan's objective. I think Kuzan affiliated himself with Blackbeard as this would give him the highest chance to reach Raftel and know the answers he is looking for. Kuzan has always been in close contact with Ds. Monkey D Garp is his idol and Jaguar D Saul was his friend. He probably knows a D would be a safer bet to reach Raftel than the likes of Kaido-Big Mom.