In Mahabharata, Shalya parva, Chapter 44.
The origins story of Skand (Lord Kartikeya), is described as follows:
In days of yore the vital seed of Maheshvara coming out, fell into a blazing fire. The consumer of everything, the adorable Agni, could not burn that indestructible seed. On the other hand, the bearer of sacrificial libations, in consequence of ...
Correct version of sharabh story is found in padma purAna, uttar khanD chapter 18:
Then the demon (jalandhar) was struck on his chest with a spear by shiv. From his mouth fearful and yawning jvar emerged. He having the face of a lion and a human figure was (called) vIrjvar.
Seeing jvar having a lion’s face, moving out of the demon’s body, (shiv) made a ‘hum’...
She is a disciple of Shiva but not his daughter.
She is said to be the mind born daughter of Sage Kashyapa in Devi Bhagavatam 9.47:
38-58. Nârâyana said :-- O Nârada! Thus I narrated to you the stories
of Sasthî and Mangala Chandîkâ, according to the Vedas. Now hear the
story of Manasâ that I heard from the mouth of Dharama Deva.
Manasâ is the mind-born ...
As per all websites or books that contain this Stotra, it was composed by Rishi Vashishta.
The Phalasruti mentions this:
Vasishtena Krutham stotram, Sarva sampathkaram param, Trisandhyam ya padenithyam, Sa hi swargamavapnyuth.
This prayer composed by sage Vasishta, Is one which gives all types of wealth, And whoever chants it at dawn, noon and dusk, Would
There are Stotras for both Lord Dakshinamurty and Nataraja.
There is the famous Dakshinamurty Stotra authored by Adi Shankaracharya.
मौनव्याख्या प्रकटित परब्रह्मतत्त्वं युवानं वर्षिष्ठांते वसद्
ऋषिगणैः आवृतं ब्रह्मनिष्ठैः । आचार्येन्द्रं करकलित
चिन्मुद्रमानंदमूर्तिं स्वात्मारामं मुदितवदनं दक्षिणामूर्तिमीडे ॥१॥
Mauna-Vyaakhyaa Prakattita Para-Brahma-Tattvam ...
There are a lot of ambiguity in regards to the origins of Lord Kartikeya or Lord Muruga. Apart from the Sanskrit literature you quoted, the Chandogya Upanishad 7.26.2 has a reference to Sanatkumara whom is regarded as Lord Kartikeya in this context.
tadeṣa śloko na paśyo mṛtyuṃ paśyati na rogaṃ nota duḥkhatāṃ sarvaṃ ha paśyaḥ paśyati sarvamāpnoti sarvaśa ...