The Emperor is not "Dead" in the true sense of the word, while his body has withered away his psychic being, powered as it is by the death of thousands of Psykers, is as strong, if not stronger, then it always was.
Not only is he able to power the Astronomican across the milky way, as has been seen in various stories and source materials he can also communicate directly with individuals if he chooses to, for instance the Inquisitor Draco, Gulliman, Alicia Dominica post resurrection etc.
In addition the general populace generally worship the Emperor as a god, therefore as far as they are aware after Horis he ascended in some form and now sits over them all watching them. It is just as likely that if they knew his body is dead they would see that as a sign of his divinity and him having ascended.
Now the second part of your question, the high lords of Terra know the Emperor is alive and kept alive by the golden throne. however beyond that it would strictly be need to know, however as explained above that isn't an issue. As long as people believe the emperor is a god who watches over Mankind and accept the Astronomican as proof of that then it is easy for the High Lords of Terra to claim there position has been decreed by god.