Is it the responsibility of the dvija castes to teach the fourth varna rigteous conduct?
Yes, Shudras would serve a Brahmana master, and in turn, the Brahmana would teach him about Dharma, God, and how to perform Pakayajnas and Sraddha.
1.103 - This [the scriptures] may be studied with care, and duly taught to pupils, by the learned Brāhmaṇa,—not by any one else.
For mere wealth and material comforts, Shudras can serve Kshatriyas and Vaishyas:
10.121 - The Śūdra, seeking a livelihood, may serve the Kṣatriya; or the Śūdra may seek to subsist by serving a wealthy Vaiśya.
But for Dharma and spiritual merit, he should serve Brahmanas:
10.122 - He shall serve Brāhmaṇas either for the sake of heaven, or for the sake of both [wealth]; when, he has attained the title of the ‘Brāhmaṇa,’ this implies the accomplishment of all his purposes.
10.123 - The service of the Brāhmaṇa is described as the distinctive duty of the Śūdra; everything else that he does is fruitless for him.
10.124 - They [Brahmanas who are served by Shudras] should provide out of their family, a suitable maintenance for him, after considering his own capacity, and the man’s skill and the burden of persons to be supported by him.
Shudra receives welfare from Brahmanas who he serves:
10.125 - Remnants of food and worn-out clothes shall be given to him; as also the grain-refuse and old furniture.
Shudras are not obligated to rites, but they have the option to if they want good results after death:
10.126 - For the Śūdra there is no sin; nor is he worthy of any sacraments; he is not entitled to any sacred rites; but there is no prohibition against sacred rites.
He can perform certain yajnas called the Pākayajñas, and also śrāddha:
Gautama (10.64-65).—‘If permission has been given to him, the Śūdra
may use the syllable namaḥ as his mantra. Some people declare that he
may himself offer the Pākayajñas.’
Gautama (4.64). - ‘He shall perform the Pākayajña rites’
Yājñavalkya (1.121).—‘Devoted to his wife, pure, supporting his
dependants, devoted to the performance of śrāddha, he should not omit
to perform the fire sacrifices with the syllable namaḥ as his mantra.’
Shudras who want to acquire lots of Punya should imitate the practice of righteous men:
10.127 - If those who, knowing their duty, and wishing to acquire merit, imitate the practices of righteous men, with the exception of reciting the sacred texts, they incur no guilt; they obtain praise.
And from other scriptures:
Viṣṇupurāṇa (Parāśaramādhava-Ācāra, p. 419).—‘It is only through
attending upon the twice-born that the Śūdra becomes entitled to
perform the Pākayajñas; and thereby becoming blessed, he wins the
worlds.—The Śūdra also shall make gifts, and perform the
Pākayajña-sacrifices, as also the rites in honour of Pitṛs.’
Mahābhārata—Anuśāsana (Do.).—‘Finding the Śūdra oppressed with bad
traits due to the quality of Tamas, Pitāmaha ordained attendance upon
the twice-born as his duty. Through his devotion to the twice-born,
the Śūdra drops off all those traits due to the quality of Tamas; and
by attending upon the twice-born, the Śūdra attains the highest
good.—Harmless, devoted to good deeds, worshipful towards gods and the
twice-born, the Śūdra becomes endowed with all the rewards of Dharma.’
Mahābhārata—Śānti (Parāśaramādhava-Ācāra, p. 420).—‘The syllables
svāhā and namaḥ are the mantras prescribed for the Śūdra; by means of
these shall the Śūdra offer the Pākayajña-Sacrifices; he shall never
amass wealth; having acquired wealth by service, he shall win the
affection of his elders; and if he is righteously inclined, he may
make gifts when permitted to do so by the King.’
Yājñavalkya (Do., p. 419).—‘Attached to his wife, pure, supporting his
dependants, devoted to the performance of Śrāddhas, he should never
omit to perform the Five Sacrifices by means of ths mantra namaḥ.’