What is the relation or connection of Saraswati (Vak - Vani) with the River Saraswati and with the Hindu Trinity or Brahman?
They are not completely same as river Saraswati is a part incarnation of Goddess Saraswati.
Devi Bhagvatam Book 9 Chapter 6 gives the reason for Ganga, Saraswati and Lakshmi (as Padmavati) to come down in the form of rivers. Due to a serious altercation between the three wives of Lord Vishnu (yes Saraswati is mentioned as Vishnu's wife before she joined Brahma), the three of them curse each other to be born as rivers. This is when Lord Vishnu calms them down and tells Lakshmi and Ganga that they will have to incarnate as rivers. Then he addresses Saraswati and says:
O Bharatî! Let you go also and incarnate in part in Bhârata under the curse of Gangâ. O Good-natured One! Now go in full Amsas to Brahmâ and become His wife. Let Gangâ go also in Her fullness to S'iva. Let Padmâ remain with Me. Padmâ is of a peaceful nature, void of anger, devoted to Me and of a Sâttvika nature.
This implies that while Saraswati in her full glory went to Brahma as his wife, a part of her incarnated on earth as the river Saraswati. This is again apparent from the many incidents that show the river goddess Saraswati to be a less powerful deity than Goddess Saraswati. One of them comes from Book 9 of Shalya Parva of Mahabharat:
Having settled this, the illustrious and great Rishi Vishvamitra with eyes red in wrath, thought of that foremost of rivers (Sarasvati). Thus remembered by the ascetic, she became exceedingly agitated. The fair lady, however, repaired to that Rishi of great energy and great wrath. Pale and trembling, Sarasvati, with joined hands appeared before that foremost of sages. Indeed, the lady was much afflicted with grief, even like a woman who has lost her mighty lord. And she said unto that best of sages, 'Tell me what is there that I shall do for thee.' Filled with rage, the ascetic said unto her, 'Bring hither Vasishtha without delay, so that I may slay him.' Hearing these words the river became agitated. With joined hands the lotus-eyed lady began to tremble exceedingly in fear like a creeper shaken by the wind.
Compare this with the following incident from the Skanda Purana in which Goddess Saraswati curses ALL the deities including the Trinity. Once when Saraswati took time to get ready for a Yagna, Brahma replaced her with Gayatri:
"At this time Savitri, accompanied by the wives of Vishnu, Rudra, and the other gods, came to the place of sacrifice. Seeing the milkmaid in the bride's bower, and the priests engaged in the performance of the sacred rites, incensed with anger, she thus addressed Pitāmāha:......'By the powers which I have obtained by the performance of tapas, may Brahmā never be worshipped in temple or sacred place, except one day in each year. . . . And, Indra, since thou didst bring that milkmaid to Brahmā, thou shalt be bound in chains by thine enemies, and confined in a strange country; and thy city and station shall be occupied by thine enemies.'
Addressing Vishnu, she said, 'Since thou gavest her in marriage to Brahmā, shalt thou, in consequence of Bhrigu's curse, be born amongst men, and shalt endure the agony of having thy wife ravished from thee by thine enemy; and long also shalt thou wander, the humble keeper of cattle!'
To Rudra she said: 'By the curse of the holy sages, shalt thou be deprived of thy manhood!' To Agni she said: 'Mayest thou be a devourer of all things, clean and unclean!' To the priests and Brāhmans she said: 'Henceforth shall ye perform sacrifices solely from the desire of obtaining the usual gifts: and from covetousness alone shall ye attend temples and holy places; satisfied only shall ye be with the food of others, and dissatisfied with that of your own houses; and in quest of riches shall ye unduly perform holy rites and ceremonies! '
This powerful Goddess who can curse even the Holy Trinity is a far cry from the feeble & trembling river-goddess who was cursed by Vishwamitra! Again, there are incidents that talk about the river-goddess marrying human kings like Ganga which is never spoken about the Goddess Saraswati. Section XCV of the Sambhava Parva of Mahabharat actually mentions her even siring a son in whose lineage Maharaj Bharat was born:
And Riksha married Jwala, the daughter of Takshaka, and he begat upon her a son of the name of Matinara, who performed on the bank of Saraswati the twelve years' sacrifice said to be so efficacious. On conclusion of the sacrifice, Saraswati appeared in person before the king and chose him for husband. And he begat upon her a son named Tansu. "Tansu was born of Saraswati by Matinara. And Tansu himself begat a son named Ilina on his wife, the princess Kalingi. "Ilina begat on his wife Rathantari five sons, of whom Dushmanta was the eldest. And Dushmanta took to wife Sakuntala, the daughter of Viswamitra. And he begat on her a son named Bharata.
The Shalya Parva of Mahabharat also mentions that she gave birth to the rishi Saraswat by association with Rishi Dadhichi. When Indra sends an Apsara called Alambusha to distract the sage his semen falls into the flowing river.
Thither where on the banks of the Sarasvati the high-souled sage was engaged in the act of gratifying the gods, the celestial damsel named above, O monarch, made her appearance. Beholding that damsel of beautiful limbs, the vital seed of that ascetic of cleansed soul came out. It fell into the Sarasvati, and the latter held it with care. Indeed, O bull among men, the River, beholding that seed, held it in her womb. In time the seed developed into a foetus and the great river held it so that it might be inspired with life as a child. When the time came, the foremost of rivers brought forth that child and then went, O lord, taking it with her, to that rishi.
Beholding that best of rishis in a conclave, Sarasvati, O monarch, while making over the child, said these words, 'O regenerate rishi, this is thy son whom I held through devotion for thee! That seed of thine which fell at sight of the apsara Alambusa, had been held by me in my womb, O regenerate rishi, through devotion for thee, well knowing that that energy of thine would never suffer destruction! Given by me, accept this faultless child of thy own!'
Thus addressed by her, the rishi accepted the child and felt great joy. Through affection, that foremost of brahmanas then smelt the head of his son and held him in a close embrace, O foremost one of Bharata's race, for some time. Gratified with the River, the great ascetic Dadhica then gave a boon to her, saying, 'The vishvadevas, the rishis, and all the tribes of the gandharvas and the apsaras, will henceforth, O blessed one, derive great happiness when oblations of thy water are presented unto them!'
Clearly, from all these passages it would seem that while Goddess Saraswati in her full potency lives with Brahma as his wife, a part of her incarnated on earth as the river-goddess Saraswati.
They're all one and the same. That is to say, Brahma's wife, who has a variety of names including Saraswati and Vak, is both the goddess of the Saraswati river and the goddess of speech/learning.
To demonstrate that Saraswati and Vak are one and the same, here is an excerpt from the Matsya Purana which describes the birth of Saraswati:
Lord Brahma, after creating His manasa putras, was not quite satisfied with the work of His creation.... After some time the goddess Gayatri, known under different names, viz. — Satarupa, Savitrf, Sarasvati, Brahmani, etc, — appeared in the form of a girl from the half portion of Brahma's body.... Afterwards, the Creator, seeing that form of exquisite beauty, was fired with love and repeatedly uttered, " What an enchanting form!" At this, the Manas putras of Brahma, i.e., Vasishtha, etc., taking Savitri for their sister, began to express their feelings of seething indignation and contempt at the attitude of Their Father (Lord Brahma), but He was so much absorbed in love that He did not heed anything in the least.
O Vidura, we have heard that Brahmā had a daughter named Vāk who was born from his body and who attracted his mind toward sex, although she was not sexually inclined towards him. Thus, finding their father so deluded in an act of immorality, the sages headed by Marīci, all sons of Brahmā, spoke [against it].
So these are just different names for the same goddess.
Now as far as the Saraswati river goes, Brahma's wife Saraswati took the form as a river as a way of dealing with the famous Vadavagni fire, as described in this excerpt and this excerpt from the Prabhasa Khanda of the Skanda Purana. Most people only know about Vishnu's incarnation Parashurama, the brahmana who killed the kings over the world 21 times because the evil king Kartavirya had Parashurama's father Jamadagni killed. But Kartavirya Arjuna and his Haihaya army had been tormenting Parashurama's tribe, the Bhargava Brahmanas, for a long time before that incident. Once, before Parashurama was even born, a Bhargava sage decided to fight back, by creating an immensely powerful fiery horse called Vadvagani.
Vadavagni was virtually uncontrollable, and it decided that it wanted to destroy the three worlds with its fire, starting with the gods. The gods turned to Brahma for help, and Brahma (as usual) turned to Vishnu. Vishnu went to Vadavagni and persuaded it that it would be impractical to kill all the gods at once, so it would be better to kill them one at a time, and he suggested that it should start by killing Varuna the ocean god. Vadavagni agreed and asked to be carried to the ocean. But none of the existing rivers were willing to carry it, because they were afraid of being consumed by its fire. So Vishnu persuaded Brahma's wife Saraswati to turn into a river in order to carry Vadavagni to the ocean, which she agreed to do after asking Brahma's permission:
Then Janardana, the Lord of Devas, spoke to Sarasvati: "0 auspicious one, you yourself go over to the west of the briny sea. If this is carried out, all the Suras will be rid of their fear. Otherwise they will be burned by the Vaclava through its refulgence. So, save Vibudhas (Suras) from this terrible fear. 0 lady of excellent waist, like a mother you alone can bestow freedom from fear on the Suras." On being told thus by Vishnu, the powerful one, she said: "I am not free. My father, the ancient one, is waiting. Being a daughter and having Vratas to be performed, I can only do what he orders. A daughter is never independent in the past, present and future. So learned men say. 0 Hari, without the specific command of my father, I will not move even a step. So, may some other means be thought of."
Having known her attitude, Vasudeva approached Pitamaha (Brahma) and said: "This is the task of Devas. Carry it out. Excepting this virgin daughter of yours, without any apparent defect, no one else can carry the Vadava fire of great prowess." On hearing the words uttered by Vispu, the Great-grandfather sniffed the head of his virgin daughter and said lovingly: "0 fair lady, do go. Save all the Devas caught in the midst of fearful danger. Take this Vadava with you and cast it into the salt sea."
So Saraswati carried it on a long journey to the ocean, and at the end Vadavagni was so grateful that it offered Saraswati for a boon. On Vishnu's advice, Saraswati asked that Vadavagni's mouth become very small, that way it would never be able to consume the whole ocean. So to this day, the fiery horse Vadavagni dwells somewhere deep within the ocean, causing small amounts of ocean water to evaporate with its flames. It is said that when Kalki comes, Vadavagni will be Kalki's horse, which he'll use to rid the world of evil as I discuss here.