1. Pets' (shaco R, ivern R, annie R) basic attacks do not stack Conqueror not even once. They do however stack it once with their ability damage (Shaco's R explosion, Ivern R's knockup, Annie R's dot area of damage). The only Pet ability that doesn't stack it, would be Morde's ghost, i think.
3. Shaco's passive makes him get 2 stacks with one hit. The same doesn't happen on Darius for example.
4. Area of damage abilities now trigger Conqueror's true damage to non-primary / closest targets with the triggering ability.
5. Illaoi's W does the same thing, but to her attack target. Maybe for the same reason her W triggers bone plating and deals reduced damage because of it with one attack?
The ability damage triggers it, and the basic attack damage gets the true damage / deals reduced damage from bone plating.
6. Sometimes Illaoi's tentacles do not give stacks. If she W-Es, one tentacle attack will give a stack. If she E-Ws, no tentacle attack gives stacks.
From a E-R-W combo she only gets 2 stacks, from the R and W, even if tentacles slammed 10 times.
7. Mordekaiser's R refreshes the stack duration without adding additional ones, but doesn't refresh Conqueror's second effect (the 10% healing + true damage one).
Because of this, his last 2 seconds won't be affected (because Conqueror finishes after the 8th second).
8. Some other dots like Gankplank and Darius' passives don't refresh Conqueror.