O Brahmana, in the golden age, Prajapati had two daughters. O sinless one, the sisters were endowed with wonderful beauty. Named Kadru and Vinata, they became the wives of Kasyapa. Kasyapa derived great pleasure from his two wedded wives and being gratified he, resembling Prajapati himself, offered to give each of them a boon.
Source : KMG Mahabharata Adi Parva Astika Parva Section XVI
Kadru asked for a thousand snakes as her offspring.
Kadru wished to have for sons a thousand snakes all of equal splendour. And Vinata wished to bring forth two sons surpassing the thousand offsprings
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of Kadru in strength, energy, size of body, and prowess. Unto Kadru her lord gave that boon about a multitude of offspring
Source : KMG Mahabharata Adi Parva Astika Parva Section XVI
Of these thousand snakes , Sesha was the eldest.
Ugrasrava (Sauti) , the son of Lomaharshana, while narrating the name of chief snakes among the thousand, to Saunaka mentioned Shesha as the eldest.
"Sesha was born first, and then Vasuki. (Then were born) Airavata, Takshaka, Karkotaka, Dhananjaya, Kalakeya, the serpent Mani, Purana, Pinjaraka, and Elapatra, Vamana, Nila, Anila, Kalmasha, Savala, Aryaka, Ugra, Kalasapotaka, Suramukha, Dadhimukha, Vimalapindaka, Apta, Karotaka, Samkha, Valisikha, Nisthanaka, Hemaguha, Nahusha, Pingala, Vahyakarna, Hastipada, Mudgarapindaka, Kamvala Aswatara, Kaliyaka, Vritta, Samvartaka, Padma, Mahapadma, Sankhamukha, Kushmandaka, Kshemaka, Pindaraka, Karavira, Pushpadanshtraka, Vilwaka,
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[paragraph continues] Vilwapandara, Mushikada, Sankhasiras, Purnabhadra, Haridraka, Aparajita, Jyotika, Srivaha, Kauravya, Dhritarashtra, Sankhapinda, Virajas, Suvahu, Salipinda, Prabhakara, Hastipinda, Pitharaka, Sumuksha, Kaunapashana, Kuthara, Kunjara, Kumuda, Kumudaksha, Tittri, Halika, Kardama, Vahumulaka, Karkara, Akarkara, Kundodara, and Mahodara.
Source : KMG Mahabharata Adi Parva Astika Parva Section XXXV
Kadru had won a wager against her sister Vinata in an unfair manner, with the help of her sons, as a result of which she had become a servant of her sister , Kadru. And the brothers of Sesha too were of jealous nature. Hence he left them and engaged himself in penance and finally pleased Brahma Deva. Then Brahma Deva appeared before him and asked him his desire.
"Sauti said, 'The illustrious Sesha amongst them, of great renown, leaving his mother practised hard penances, living upon air and rigidly observing his vows. He practised these ascetic devotions, repairing to Gandhamadana, Vadri, Gokarna, the woods of Pushkara, and the foot of Himavat. And he passed his days in those sacred regions, some of which were sacred for their water and others for their soil in the rigid observance of his vows, with singleness of aim, and his passions under complete control. And the Grandsire of all, Brahma, saw that ascetic with knotted hair, clad in rags, and his flesh, skin, and sinews dried up owing to the hard penances he was practising. And the Grandsire addressing him, that penance-practising one of great fortitude, said, 'What is that thorn doest, O Sesha? Let the welfare of the creatures of the worlds also engage thy thoughts. O sinless one, thou art afflicting all creatures by thy hard penances. O Sesha, tell me the desire implanted in thy breast.'
Source : KMG Mahabharata Adi Parva Astika Parva Section XXXVI
Then Sesha told him about the poor treatement of Vinata and Garuda at the hands of her mother Kadru and he also told about the envious desposition of his brothers.
"And Sesha replied, 'My uterine brothers are all of wicked hearts. I do not desire to live amongst them. Let this be sanctioned by thee. Like enemies they are always jealous of one another. I am, therefore, engaged in ascetic devotions. I will not see them even. They never show any kindness for Vinata and her son. Indeed, Vinata's son capable of ranging through the skies, is another brother of ours. They always envy him. And he, too, is much stronger owing to the bestowal of that boon by our father, the high-souled Kasyapa. For these, I engaged in ascetic penances, and I will cast off this body of mine, so that I may avoid companionship with them, even in another state of life.'
Source : KMG Mahabharata Adi Parva Astika Parva Section XXXVI
The highlighted text shows that Gaurda and Sesha are not enemies.
Then Brahma Deva was impressed seeing the virtuous nature of Sesha and offered him a boon.
"Unto Sesha who had said so, the Grandsire said, 'O Sesha, I know the behaviour of all thy brothers and their great danger owing to their offence against their mother. But O Snake, a remedy (for this) hath been provided by me even beforehand. It behoveth thee not to grieve for thy brothers. O Sesha, ask of me the boon thou desirest. I have been highly gratified with thee and I will grant thee today a boon. O best of snakes, it is fortunate that thy heart hath been set on virtue. Let thy heart be more and more firmly set on virtue.'
Source : KMG Mahabharata Adi Parva Astika Parva Section XXXVI
Sesha asked Brahma Deva to grant him a boon such that his heart would always remain inclined to virtues. Very similar to the boon asked by Vibhishana.
"Then Sesha replied, 'O divine Grandsire, this is the boon desired by me; viz., may my heart always delight in virtue and in blessed ascetic penances, O Lord of all!'
Source : KMG Mahabharata Adi Parva Astika Parva Section XXXVI.
Then Brahma Deva asked him to bear the earth.
"Brahman said, 'O Sesha, I am exceedingly gratified with this thy self-denial and love of peace. But, at my command, let this act be done by thee for the good of my creatures. Bear thou, O Sesha, properly and well this Earth so unsteady with her mountains and forests, her seas and towns and retreats, so that she may be steady.'
Source : KMG Mahabharata Adi Parva Astika Parva Section XXXVI.
Hence Ananta Shesha did as commanded.
"Sesha said, 'O divine Lord of all creatures, O bestower of boons, O lord of the Earth, lord of every created thing, lord of the universe, I will, even as thou sayest hold the Earth steady. Therefore, O lord of all creatures, place her on my head.
Source : KMG Mahabharata Adi Parva Astika Parva Section XXXVI.
Ananta Shesh Naga had also played an important role in Samudra Manthan by helping the Gods in disloding Madrachal , which was used to churn the ocean.
The Gods had tried earlier but had failed in dislodging Madrachal.
"Sauti said, 'There is a mountain called Mandara adorned with cloud-like peaks. It is the best of mountains, and is covered all over with intertwining herbs. There countless birds pour forth their melodies, and beasts of prey roam about. The gods, the Apsaras and the Kinnaras visit the place. Upwards it rises eleven thousand yojanas, and descends downwards as much. The gods wanted to tear it up and use it as a churning rod but failing to do so same to Vishnu and Brahman who were sitting together, and said unto them, 'Devise some efficient scheme, consider, ye gods, how Mandara may be dislodged for our good.'
Source : KMG Mahabharata Adi Parva Astika Parva Section XVIII
Later when asked by Brahma Deva and Bhagwan Vishnu, Ananta Sesha did it single-handedly.
"Sauti continued, 'O son of Bhrigu! Vishnu with Brahman assented to it. And the lotus-eyed one (Vishnu) laid the hard task on the mighty Ananta, the prince of snakes. The powerful Ananta, directed thereto both by Brahman and Narayana, O Brahmana, tore up the mountain with the woods thereon and with the denizens of those woods. And the gods came to the shore of the Ocean with Ananta and addressed the Ocean, saying, 'O Ocean; we have come to churn thy waters for obtaining nectar.' And the Ocean replied, 'Be it so, as I shall not go without a share of it. I am able to bear the prodigious agitation of my waters set up by the mountain.' The gods then went to the king of tortoises and said to him, 'O Tortoise-king, thou wilt have to hold the mountain on thy back!' The Tortoise-king agreed, and Indra contrived to place the mountain on the former's back.
Source : KMG Mahabharata Adi Parva Astika Parva Section XVIII.
The strength of Ananta Sesh Naga knows no bounds. He has enough strength to lift the entire cosmos with its mobile and immobile creatures. This is confirmed by Brahma Deva in both Ramayana and Mahabharata.
(i) Quote from Mahabharata : Brahma Deva praised Sesh Naga as the best among snakes , and accepted his strength and ability of bearing the weight of the three worlds.
"Brahman said, 'O Sesha, O best of snakes, thou art the god Dharma, because alone, with thy huge body, thou supportest the Earth with everything on her, even as I myself, or Valavit (Indra), can.'
Source : KMG Mahabharata Adi Parva Astika Parva Section XXXVI.
(ii) Quote from Ramayana : Here Brahma Deva called Ananta Shesh Nag as an incarnation of Bhagwan Vishnu and accepted its ability of bearing the weight of all the three worlds.
"With thousand feet, with hundred heads and with thousand eyes along with Lakshmi the , goddess of wealth, you bear the earth with all its created beings along with its mountains."
"O Rama! You appear as Sesha, a large serpent in water, at the earth's bottom, bearing the three worlds, gods, Gandharvas, the celestial musicians and the demons."
Source : Valmiki Ramayana Yuddha Kanda Sarga 117.
In Anusasana Parva of Mahabharata , Bhagwan Shiv himself praises Shesh Naga and says that even Garuda could’nt find his end. He is capable of holding the weight of all the three worlds and called him an incarnation of Bhagwan Vishnu. I think the above and this quote should answer the second part of him being bed of Bhagwan Vishnu.
He is called Ananta (Infinite) Verily, that illustrious one is identical with the immutable Hari. Once on a time the mighty Garuda, the son of Kasyapa, was addressed by the deities in these words, 'Do thou, O puissant one, see if this one has any end!' Though possessed of great energy and might, Garuda, however, failed to find out the end of this illustrious one who is identical with the Supreme Soul. Supporting the whole earth on his head, he resides in the nether regions. He roves through the universe as Sesha, filled with great joy. He is Vishnu, He is the illustrious Ananta. He is the supporter of the earth.
Source : KMG Mahabharata Anusasana Parva Section CXLVII
He had incarnated as Lakshman ji to help Bhagwan Sri Ram and as Balram ji to help Bhagwan Sri Krishna in the establishment of Dharma.