Here is a partial answer based on the Devi Bhagavata Purana,BOOK 3 Chapter 26.
Maharshi Vyasa explains the significance of performing the Asvin and Chaitra Navaratris to King Janamejaya :
Janamejaya said :– “O Best of the Brâhmins! What are men to do in the time of Navarâtra? Especially in the Navarâtra ceremony during the autumnal season how is the ceremony to be performed? Kindly relate all this with the prescribed rules and regulations.
O intelligent one! What are the fruits therein of the Navarâtra ceremony? and what are the rules to be observed? Kindly describe all these to me.”
3-5. Vyâsa said :– O king! Hear about the vow of auspicious Navarâtra. This has to be performed with loving devotion in the vernal season; but its special season is autumn. The two seasons, autumn and spring, are famous as the teeth of Yama, the God of Death; and these are the two seasons, very hard for the persons to cross over. Therefore every goodfaring man should everywhere perform this vow very carefully.
6-8. O king! The people are very much afflicted with various terrible diseases in these two seasons autumn and spring and many lose their lives during these portions of the year. Therefore the wise should unquestionably worship with great devotion the Chandikâ Devî in these auspicious months of Chaitra and Âs’vin.
द्वाव्रृत्तयमदंष्ट्राख्यौ नूनं सर्वजनेषु वै। शरदः सन्तानमनौ दुर्गमौ प्राणिनामिह॥ तस्मात् यत्नतः इदम् कार्यं सर्वत्र शुभमिच्छुना॥(देवीभागवतम्, तृतीयस्कन्धम्, २६ अध्यायः)
Yama is son of Surya. He is called as Dharmaraja. He will not go astray from the righteous path. He condemned the folly done by Chaya Devi, second wife of his father. He is an embodiment of peace for the good. He is a tyrant for the sinners. He has two long and sharp teeth on both sides of his mouth. They are the VASANT and SHARAD RITUS . (Duration of every two months is called a RITU. They are six Ritus: VASANTA, GREESHMA, VARSHA, SHARAD, HEMANTA & SHISHIRA). All the diseases under the sun are YAMA’s servants to fetch the lives of those dead in this world. He is by himself very ferocious in appearance. He is also known as BHAYANKARA. Those who see his long teeth protruding on the sides of the mouth are sure to shudder. In the VASANTH and SHARAD RITUS, the world suffers various diseases
To escape from the clutches of Yama's two teeth the Sharad and Vasant(Chaitra) Navaratris are celebrated.