Kausalyā’s husband was alive. [Therefore, it isn’t appropriate for her to leave her husband and home to live in the forest with her son to take up the austerity of vānaprastha life.]

Upto this point, Rāma does not know that Sītā would [adamantly] want to accompany Him to the forest. [But] He has already heard Lakṣmaṇa tell Queen Kausalyā that He would accompany Rāmacandra to the forest:

dīptam agnim araṇyaṁ vā yadi rāmaḥ pravekṣyati
praviṣṭaṁ tatra māṁ devi tvaṁ pūrvam upadhāraya

“O queen, know [for sure that] if Rāma enters the blazing fire or a forest, I would have entered them before Him.” (Rāmāyaṇa 2.21.18)

So “Please pray for Our welfare” here means “Please pray for My and Lakṣmaṇa’s welfare.” Therefore, Lakṣmaṇa tells Rāma later on that Rāma had already allowed Lakṣmaṇa to come with Him—anujñātaś ca bhavatā pūrvam eva (Rāmāyaṇa 2.31.7).

The history of Yayāti is described in the Mahābhārata. He had fallen from Svarga [but] he returned to Svarga because of his truthfulness.

NOTE. A summary of Yayāti’s history from the Ādi-parva of Mahābhārata is as follows.

In the final years of his life, Yayāti retired from ruling his kingdom and departed to the forest as a vānaprastha. By dint of his austerities, he left his body and ascended to Svarga where he lived for a long time.

One day, he met Indra, who asked him who was as good as him in austerity. Yayāti replied that he hadn’t seen anyone among the devas, gandharvas, great sages and human beings whose austerities were as good as his. Indra then told him that Yayāti was too proud of himself and disregarded the accomplishments of several ascetics, and as as a result of his prideful answer to his question, he had lost his remaining piety that was required for him to stay in Svarga. Yayāti then requested that he fall into a place populated by virtuous sages. Indra agreed and advised Yayāti to be respectful to others’ religious accomplishments in austerity.

When Yayāti was falling from the heavens, four saintly kings engaged in austerities noticed him in the sky. They were Aṣṭaka, Pratardana, Śibi and Vasumanā—born in Yayāti’s lineage. They asked him about him, his whereabouts and then asked him about his experience in Svarga, loss of piety and related topics. Yayāti answered his questions and the the saintly kings prevented Yayāti from falling further using their mystic powers gained from their performances of austerity.

They offered to transfer their pious credits to him so that he could return to Svarga. But Yayāti declined each of their offers, saying that he wanted to subsist on the pious credits that he had earned himself. When Vasumanā suggested that Yayāti could give him some grass in exchange for his pious credits—this would no longer be a charity of pious credits—Yayāti still refused, saying that that would be falsehood on his part. Soon, five golden aerial chariots descended from the skies to take Yayati and his four grandsons to heaven. Yayāti thus returned to Svarga because of his adherence to truthfulness.