Was the name "Ramayana" given to Ramayana by sage Valmiki or is it people who just call it by that name?
- If not, then what was the name given by Valmiki?
- Is the Ramayana book written by the sage Valmiki still available?
- If not, then what is the oldest book/scriptures/source of Ramayanam?
- Was it divided into 5 or 7 kandas by Valmiki himself?
- Also, was it Valmiki himself, who gave the names/titles to the volumes (e.g., "Bala Kanda")?
Similarly, was the name "Mahabharata" given by sage Krishna Dwaipayana (Veda Vyasa)?
- Was it divided and named as the various parvas by the sage himself?
- What's the oldest source of the Mahabharata?
- I know that Bhagavad Gita is a part of the Mahabharata. But was the name 'Bhagavad Gita' given by Veda Vyasa?
- If not, then how did the name 'Bhagavad Gita' come into usage?
Let me discuss each of the three names separately:
Ramayana: As described in this chapter of the Bala Kanda of the Ramayana, after Valmiki composed his famous epic poem (a story I discuss here), he gave it three names: Ramayana, Sitayascharitam Mahat, and Paulastya Vadha:
That sage with observed sacred vows has rendered the entire epic in the name of 'Ramayana', 'Sublime Legend of Seetha' and 'elimination of Ravana'. [1-4-7]
Mahabharata: As described in the first chapter of the Adi Parva of the Mahabharata, when Vyasa asks Ganesha to write down the Mahabharata for him, he calls it simply the Bharata:
Then Vyasa began to call to mind Ganesa. And Ganesa, obviator of obstacles, ready to fulfil the desires of his votaries, was no sooner thought of, than he repaired to the place where Vyasa was seated. And when he had been saluted, and was seated, Vyasa addressed him thus, 'O guide of the Ganas! be thou the writer of the Bharata which I have formed in my imagination, and which I am about to repeat."
How it got the appellation "Mahabharata" is that it was found to be "heavier" than the Vedas, as described later on in the same chapter:
In former days, having placed the four Vedas on one side and the Bharata on the other, these were weighed in the balance by the celestials assembled for that purpose. And as the latter weighed heavier than the four Vedas with their mysteries, from that period it hath been called in the world Mahabharata (the great Bharata). Being esteemed superior both in substance and gravity of import it is denominated Mahabharata on account of such substance and gravity of import. He that knoweth its meaning is saved from all his sins.
Bhagavad Gita: Vyasa's disciple Vaishampayana calls Krishna's discourse to Arjuna the "Hari Gita" in this chapter of the Narayaniya section of the Shanti Parva of the Mahabharata:
In the beginning of the Treta Yuga, Vivaswat imparted the knowledge of this cult to Manu. Manu, for the protection and support of all the worlds, then gave it to his son Ikshaku. 2 Promulgated by Ikshaku, that cult over-spreads the whole world. When the universal destruction comes, it will once more return to Narayana and be merged in Him. The religion which is followed and practised by the Yatis, has, O best of kings, been narrated to thee before this in the Hari Gita, with all its ordinances in brief.
That's just a synonym for the Bhagavad Gita, because both Bhagavan and Hari are terms for Vishnu. In any case, the title of the section of the Bhishma Parva of the Mahabharata where the Bhagavad Gita occurs is the Bhagavad Gita Parva; see here.
You also wanted to know about original texts. Well, despite Ganesha writing down the Mahabharata, writing is not the primary way the Ramayana and Mahabharata were passed down; like the Vedas they were initially passed down via oral tradition, and only passed down in written form much later. So we don't really have "original texts" in that sense. But if you want to read them in the original Sanskrit, you can read the Ramayana here and the Mahabharata here.