First of all, let me make it clear that Lord Vedavyasa is not just immortal but Bhagavan Vishnu Himself, who incarnated in the 28th Dvapara yuga and took the Vyasa position. So yes, coming to the first part of the question. No one writes the Vedas. They are Apaurusheya, which means they are not composed by anyone. They are eternal and always praise Bhagavan.
अस्य महतो भूतस्य निःश्वसितमेतद्यदृग्वेदो यजुर्वेदः सामवेदोऽथर्वाङ्गिरस इतिहासः पुराणं विद्या उपनिषदः श्लोकाः सूत्राण्यनुव्याख्यानानि व्याख्यानानि। अस्यैवैतानि सर्वाणि निःश्वसितानि॥
The Rig Veda, Yajur Veda, Sama Veda, and Atharvaveda, history, puranas, arts, upanishads, verses, aphorisms, elucidations, and explanations are all the breath of this Great Reality (Brahman). From It indeed are all these breathed forth.
[Brihadaranyaka Upanishad (2.4.10)]
अपौरुषेय-वाक्य-अङ्गीकारे न किञ्चित् कल्प्यम् । अपौरुषेयत्वं च स्वतः एव सिद्धम् ॥
In accepting the concept of an unauthored statement (the Veda), no arbitrary assumptions are needed. Furthermore, its unauthored nature (Apaurusheyatva) is self-evident (Svatah Siddham).
[SriVishnu Tattva Vinirnaya by SriMadhvacharya]
तस्माद्यज्ञात्सर्वहुत ऋचः सामानि जज्ञिरे । छन्दांसि जज्ञिरे तस्माद्यजुस्तस्मादजायत ॥
From that ultimate cosmic sacrifice (of the Supreme Purusha), the Richas (Rig Veda) and the Samans (Sama Veda) were born. From That the poetic meters were born, and from That the Yajus (Yajur Veda) was born.
[Purusha Sukta, Rigveda]
Plus, the Vedas are flawless and don’t have any contradictions. That’s why they are placed at a higher authority when the Purana statements contradict each other. If a Purana statement is against the Vedas, Upanishads, Brahma Sutras, and the Bhagavad Gita, it must be ignored, because that statement is kept there for delusion, to separate the ignorant people from the devoted noble people. Hence, the Vedas are not composed by anyone and hence are Apaurusheya.
श्रुतिस्मृतिपुराणानां विरोधो यत्र दृश्यते ।
तत्र श्रौतं प्रमाणं तु तयोर्द्वैधे स्मृतिर्वरा ॥
Wherever a contradiction is seen between the Shruti (Vedas), Smriti, and Puranas, the statement of the Veda (Shruti) must be accepted as the ultimate authority. If there is a conflict between Smriti and Puranas, the Smriti is superior.
[Vyasa Smriti]
या वेदबाह्याः स्मृतयो याश्च काश्च कुदृष्टयः ।
सर्वास्ता निष्फलाः प्रेत्य तमोनिष्ठा हि ताः स्मृताः ॥ ९५ ॥
"All those traditions (Smritis) that are outside the fold of the Veda, and all those despicable systems of philosophy based on distorted logic (Kudrishti), are completely useless and produce no reward after death; for they are explicitly declared to be founded upon the mode of darkness (Tamas).
[Manu Smriti]
Vedas, being eternal, were divided into parts earlier by Lord Vishnu Himself in the form of Hayagriva. But still, people couldn’t understand the correct meaning and purport of the Vedas. Hence, that same Bhagavan Vishnu incarnated as Lord Vedavyasa and divided the Vedas into 4 parts, namely Rik, Yajus, Sama and Atharvaveda respectively. That’s why He is known as Vyasa. But still, there were some places where there were possibilities to have contradictions in the Vedas if not understood properly. That’s why Lord Vedavyasa wrote the Brahma Sutras. The Brahma Sutras contain small formula-like statements, clarifying the true meaning of the Vedas.
ऋषिभिर्बहुधा गीतं छन्दोभिर्विविधैः पृथक् ।
ब्रह्मसूत्रपदैश्चैव हेतुमद्भिर्विनिश्चितैः ॥ ५ ॥
This truth has been sung by the great Rishis in many different ways, in various distinct Vedic hymns (Chandas), and especially stated in the aphorisms of the Brahma Sutras, which are structured with flawless logic (Hetumadbhiḥ) and are completely decisive (Viniścitaiḥ).
[BhagavadGita, Adhyaya 13, Shloka 5]
Lord Vedavyasa, after dividing the eternal Vedas, composed the Mahabharata, and then took part Himself along with Krishna in the Mahabharata. The Puranas are NOT MAN-WRITTEN. Lord Vedavyasa compiled the Purana Samhita and divided it into the eighteen Mahapuranas
चातुर्होत्रं व्यवस्थानं राद्न्यां यज्ञसतत्ये ।
वेदमेकं चतुर्धा व्यभजत् पुरुषर्षभः ॥ १९ ॥
इतिहासपुराणं च पञ्चमो वेद उच्यते ॥ २० ॥
That greatest of men (Vyasadeva) divided the one, single Veda into four distinct divisions to ensure the continuation of sacrifices. Furthermore, the Itihasas (histories like Mahabharata) and the Puranas are declared to be the fifth Veda." (The subsequent verses name his disciples, confirming he handed the compiled Puranas over to Romaharsana).
[SrimadBhagavatam 1.4.19–20]
आख्यानैरप्युपाख्यानैर्गाथाभिः कल्पशुद्धिभिः ।
पुराणसंहितां चक्रे पुराणार्थविशारदः ॥ ४ ॥
प्रख्यातो व्यासशिष्योऽभूत् सूतो वै रोमहर्षणः ।
पुराणसंहितां तस्मै ददौ व्यासो महामुनिः ॥ ५ ॥
The great expert in the meaning of ancient lore (Vyasadeva) compiled the Purana Samhita using core narratives (Akhyanas), secondary episodes (Upakhyanas), allegorical verses, and chronicles of various cosmic cycles (Kalpas). The great sage Vyasa then handed over this compiled Purana Samhita to his renowned disciple, the Sutta named Romaharsana.
[Vishnu Purana 3.6.4–5]
So, in the case of the Vedas, Lord Vedavyasa divided them into 4 Vedas and composed the Brahma Sutras. He even composed the Mahabharata, which is the Itihasa, and then divided the Purana Samhita into 18 Mahapuranas. How could He write so much? Because He was none other than Bhagavan Vishnu Himself, who incarnated to spread true knowledge among people. He is non-different from Bhagavan Vishnu, and nothing is impossible for Bhagavan Vishnu. Hence, He could write these granthas like the Mahabharata and the 18 Puranas.
व्यासाय विष्णुरूपाय व्यासरूपाय विष्णवे ।
नमो वै ब्रह्मनिधये वासिष्ठाय नमो नमः ॥ ४ ॥
Salutations to Vyasa, who is the form of Vishnu, Vedavyasa roopi Bhagavan Vishnu. Repeated salutations to him who is a treasure house of Vedic knowledge, and who is a descendant of the lineage of Sage Vashishta.
[Mahabharata, Vishnu Sahasranama]
द्वापरे द्वापरे विष्णुर्व्यासरूपी महामुने ।
वेदमेकं बहुधा कुरुते जगतो हितम् ॥
O great sage! In every Dvapara Yuga, Lord Vishnu Himself takes the form of Vyasadeva. For the ultimate benefit of the world, He takes the single, unified Veda and divides it into multiple parts.
[Vishnu Purana, 3.3.11–12]
ततः सप्तदशे जातः सत्यवत्यां पराशरात् ।
चक्रे वेदतरोः शाखा दृष्ट्वा पुंसोऽल्पमेधसः ॥ २१ ॥
Thereafter, in the seventeenth incarnation of Godhead, the Lord appeared in the womb of Satyavati, born from Sage Parashara... Seeing that humanity in general would become less intelligent, He divided the desire-tree of the Veda into various branches.
[SrimadBhagavatam 1.3.21]
Why could He compose all these works? Because He is Bhagavan Vishnu Himself, who is the extolled deity in the Vedas. Therefore, Lord Vedavyasa composed all these great works, showing His eternal and pure mercy upon us.
कृष्णद्वैपायनं विद्धि नारायणमीश्वरम् ।
को ह्यन्यः पुण्डरीकाक्षाद् महाभारतकृद् भवेत् ॥
Know that Sage Krishna Dvaipayana (Vedavyasa) is none other than the Supreme Lord Narayana Himself. For who else on this earth, other than the lotus-eyed Lord Vishnu, could have composed the magnificent epic Mahabharata?
[Vishnu Purana 3.4.5]
Hope this helps!
Jay SriRam!
ShriKrishna Dvaipayana Vedavyasa Abhinna ShriKrishnarpanamastu!