Omniscience means He should know everything including all our actions and whatever we talk. Is this correct?
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Yes God knows even what you talk secretly. See the following Atharva Veda (AV) Mantra:
Dvau samnishadya yat mantrayete rAja tat veda varunah tritiyah ||
When you talk in secret, there is always the third observer, divine observer Varuna.
So, God is monitoring all our actions, even though we might think that we are doing those actions secretly.
The following three Manu Smriti verses make things even more clear.
8.84. ’The Soul itself is the witness of the Soul, and the Soul is the refuge of the Soul; despise not thy own Soul, the supreme witness of men.
8.85. ’The wicked, indeed, say in their hearts, "Nobody sees us;" but the gods distinctly see them and the male within their own breasts.
8.86. ’The sky, the earth, the waters, (the male in) the heart, the moon, the sun, the fire, Yama and the wind, the night, the two twilights, and justice know the conduct of all corporeal beings.’
So, yes Gods very well know what we are doing or talking although we might be thinking that we have done those actions "secretly".
Yes, He knows everything, each and every detail of each n every soul. References from our scriptures have already been stated above in one of the answers by Rickross, but just adding something more to that.
Each and every minute details be it words, action, thoughts etc of each n every soul is captured in Akashic Records, the Akashic records are a compendium of all human events, thoughts, words, emotions, and intent ever to have occurred in the past, present, or future.
Citing references from Wiki:
In theosophy and anthroposophy, the Akashic records are a compendium of all human events, thoughts, words, emotions, and intent ever to have occurred in the past, present, or future. They are believed by theosophists to be encoded in a non-physical plane of existence known as the etheric plane. There are anecdotal accounts but there is no scientific evidence for the existence of the Akashic records.13
Akasha (ākāśa आकाश) is the Sanskrit word for 'aether' or 'atmosphere'.
Also as described Akashic records site here:
The Akashic Records are the energetic records of all souls about their past lives, the present lives, and possible future lives. Each soul has its Akashic Records, like a series of books with each book representing one lifetime. The Hall (or Library) of the Akashic Records is where all souls’ Akashic Records are stored energetically. In other words, the information is stored in the Akashic field (also called zero point field). The Akashic Records, however, are not a dry compilation of events. They also contain our collective wisdom.
In Hinduism we do refer to Chitragupta as a person who keeps track of each & every action of everyone.
Chitragupta (Sanskrit: चित्रगुप्त, 'rich in secrets' or 'hidden picture') is a Hindu god assigned with the task of keeping complete records of actions of human beings on the earth. He is god of justice. Upon their death, Chitragupta has the task of deciding heaven or the hell for the humans, depending on their actions on the earth. Chitragupta Maharaj (Chitragupta ) is the patron deity of Kayasthas, a Hindu caste of India and Nepal.
More can be read about him in the following question: Is Chitragupt of Yamlok mentioned in any scriptures?
God doesn't know about your thoughts, only you are aware about it. Sri Yukteswar told about the workings of thought reading to his disciple Paramahansa Yogananda.
Autobiography of a Yogi / Chapter 12 - Years in My Master’s Hermitage:
Unless necessary I do not invade the seclusion of others’ minds. Man has the natural privilege of roaming secretly among his thoughts. The unbidden Lord does not enter there; neither do I venture intrusion.
Similar to above, Bhagavad Gita also suggests that, the supreme One 'acts' disinterested and secluded to the worldly matters:
BG 9.9 - O Arjuna, those actions (Karma-s) don't bind Me, as "I" am indifferent (disinterested) & detached to those actions
BG 13.15 - Reflecting through the senses, [yet] devoid of all the senses; disinterested; the sustainer of all; Without 3 modes, and the consumer of 3 modes
To be able to "know" anyone's thoughts, mind & senses are required. However, the omnipresent is devoid of all senses. Hence, it merely acts as a witness.