Shishui and Itachi had strong connection with the ROOT program, in fact, it seems like they were part of Danzo's ROOT program. Not to mention that Tobi and Danzo said that they hadn't seen each other since the Massacre, which implies that Danzo was in the Uchiha compound during the Massacre.
Now let us face some facts, Itachi, even with Tobi's help would have never been able to kill the whole clan without using MS, it is simply impossible, that is common sense.
Now we have Itachi and Shishui as part of ROOt, and Danzo being involved during the massacre. It becomes obvious at this point how and why Danzo knows what he knows about the Uchiha.
Also, he would have been able to read the tablet after he obtained Shishui's eye, which happened before the massacre.
That is how Danzo would have known.
Hiruzen would not have known simply because Danzo knew, because Danzo plotted behind Hiruzen's back on many occassions, he even teamed up with Orochimaru so that he can become Hokage after Oro kills Hiruzen. And did the same thing to Tsunade when he allowed Pein to destroy the village while he hid under ground.
This person, judging by his character, would not have shared anything with Hiruzen.
However, as Hokage, he might have had access to some info that Hashirama and Tobirama passed on, since Minato spoke of Madara as if he had intel when he fought Tobi, even using Madara's powers and character to judge how Tobi would act and attack. He surely got this info from Hiruzen, unless someone else told him, definitely not Jiraiya though, that guy didn't know shit!