(1) After burial, it is mustahab to stand at the head of the grave and recite from the beginning of Sura-al-Baqara until “مفلحون”, and then at the foot of the grave and recite from “آمن الرسول” until the end of Sura-al-Baqara. This has been established in Ahadith.
(2) Yes, this is an authentic narration, and has been narrated in many books of Ahadith.
(3) Allah is indeed omnipresent, which is a quality specific to him. However, his existence is not like that of mankind, which can be perceived by everyone.
And Allah knows best.