Who is mother Durga, why was she prayed by all gods including Vishnu to be reborn as Parvathi?
Since you asked Was Sati also a reincarnation of mother Durga ?, you should also understand why Durga Devi was born as Sati Devi.
When Brahma created beings initially, they couldn't multiply or reproduce sexually. Lord Brahma wanted female energy or Shakti to make sexual reproduction possible (so that beings would be able to reproduce) and did Tapasya for Parameswara (Shiva with Shakti) and Lord Shiva appeared before him in Ardhanareswara form and gave female energy (shakti) needed for creation.
This is described in Chapter 3, Satarudra Samhita of Shiva Mahapurana.
यदा सृष्टाः प्रजाः सर्वा न व्यवर्द्धन्त वेधसा।
तदा चिन्ताकुलोऽभूत्स तेन दुःखेन दुःखितः ॥ २॥
When there had been little evolution of the people created by Brahma, he felt extremely painful.
नभोवाणी तदाऽभूद्वै सृष्टिं मिथुनजां कुरु।
तच्छ्रुत्वा मैथुनीं सृष्टिं ब्रह्मा कर्तुममन्यत॥ ३॥
Then there was a divine sound from the sky said, "Carry on the creation produced by the couples' (i.e. the male and the female). Then Brahma thought of the creation based on the sexual relations.
नारीणां कुलमीशानान्निर्गतं न पुरा यतः।
ततो मैथुनजां सृष्टिं कर्तुं शेके न पद्मभूः॥४॥
Earlier to that the race had not appeared out of the sexual intercourse with the females. Therefore Brahma could not create the world based on sexual relationship earlier.
प्रभावेणा विना शम्भोर्न जायेरन्त्रिमाः प्रजाः।
एवं सञ्चिन्तयन्ब्रह्मा तपः कर्तुं प्रचक्रमे॥ ५॥
Brahma thought that the creation cannot be progressed without the influence of Šiva. Therefore he started performing tapas.
शिवाय परया शक्त्या संयुक्त परमेश्वरम्।
सञ्चिन्त्य हृदये प्रीत्या तपेशं परमं तपः।। ६॥
Then he meditated upon Siva united with the great Sakti called Parameswara (Siva) in a combined way performing severe tapas.
तीव्रेण तपसा तस्य संयुक्तस्य स्वयम्भुवः।
अचिरेणैव कालेन तुतोष स शिवो द्रुतम्॥७॥
Observing the severe austerities of Brahma, lord Siva was rapidly pleased with him within no time.
ततः पूर्णचिदीशस्य मूर्तिमाविश्य कामदाम्।
अर्द्धनारीनरो भूत्वा ततो ब्रह्मान्तिकं हरः॥८॥
Then Isvara- the form of complete consciousness, entered into the Ardhanarisvara form, representing half male and half female and he appeared before Brahmā.
Brahma eulogized Him and Shiva gave His Shakti to Brahma.
इत्युक्त्वा परमोदारं स्वभावमधुरं वचः।
पृथक्चकार वपुषो भागाद्देवीं शिवां शिवः॥ १३॥
Thus Siva extremely large hearted, and Sweet tempered one, separated his body from that of Siva.
तां दृष्ट्रा परमां शक्ति पृथग्भूतां शिवागताम्।
प्रणिपत्य विनीतात्मा प्रार्थयामास तां विधिः॥ १४॥
Looking at the Supreme Sakti, having been separated from the body of Siva, Brahma offered his salutation to her and then eulogised her.
Brahma then eulogized Adi Shakthi, the source of all Shaktis (female energy) in cosmos. After Brahma's prayer, Shiva gave Her Shakti to Brahma and said She would born as daughter of Daksha Prajapati (as Sati Devi).
तपसाराधिता देवि! ब्रह्मणा परमेठिना।
प्रसन्ना भव सुप्रीत्या कुरु तस्याखिलेप्सितम्॥ २६॥
Siva said, “Brahma has adored you (Shakti or Sivaa) performing the tapas. Therefore getting pleased, you better fulfil his desires.”
तामाज्ञां परमेशस्य शिरसा प्रतिगृह्य सा।
ब्रह्मणो वचनाद्देवी दक्षस्य दुहिताऽभवत्॥ २७॥
The goddess Paramesvari, accepting his command, according to the words of Brahma, she became the daughter of Daksa Prajapati.
दत्त्वैवमतुलां शक्तिं ब्रह्मणे सा शिवा मुने!।
विवेश देहं शम्भोर्हि शम्भुश्चान्तर्दधे प्रभुः॥ २८॥
O Sage, then Sivaa (Shakti), giving the unique Śakti to Brahma, entered the body of Siva, who disappeared from the scene.
तदाप्रभृति लोकेऽस्मिन्स्त्रिया भागः प्रकल्पितः।
आनन्दं प्राप स विधिः सृष्टिर्जाता च मैथुनी॥ २९ ।।
From that day onwards the share of the Woman, started to be given separately. Brahma then felt delighted and the creation based on sexual union started.
Lord Sada Shiva then took as Saguna form of Rudra (who dwells on Kailasa) and Durga or Adi Shakti (source of all Shaktis) born as Sati and Parvati in Saguna forms in different kalpas. The invisible form of Shakti (which mingled with Shiva) was born as Sati Devi first and then as Parvati and again will be born as Sati and this cycle repeats, which you can read in this answer in detail.
Brahma prayed to Durga to born as shakti in Saguna form for Rudra.
Chapter 10 Rudra Samhita (Sati Khanda) of Shiva Mahapurana says:
मद्रूपं परमं विष्णो ! ईदृशं ह्यङ्गतो विधेः।
प्रकटीभविता लोके नाम्ना रुद्रः प्रकीर्तितः।।५४।।
"Visnu is my form. Similarly Rudra would be manifested from Brahma and shall be well known in the world by the same name. He will be my form.
पूर्णरूपः स मे पूज्यः सदा वां सर्वकामकृत्।
लयकर्ता गुणाध्यक्षो निर्विशेषः सुयोगकृत्।॥५५॥
He is my complete form and lord of all the gunas and is worthy of being Worshipped by you. He will fulfil all your desires.
त्रिदेवा अपि मे रूपं हरः पूर्णो विशेषतः।
उमाया अपि रूपाणि भविष्यन्ति त्रिधा सुतौ।५६।
Brahmā, Visnu and Siva are my forms. Hara is especially my complete form. Similarly Uma shall also have three forms.
लक्ष्मीनर्गम हरे: पत्री ब्रह्मपत्री सरस्वती।
पूर्णरूपा सती नाम रुद्रपत्नी भविष्यति।॥५७॥
As Laksmi, she would be the spouse of Visnu, as Sarasvati, the spouse of Brahma, and as Rudra Patni (as Sati)- having the complete form, would be the spouse of Rudra.”
Daksha, at command of Brahma, prayed to Durga and got Her as daughter. Chapter 11 Rudra Samhita (Sati Khanda) says:
यदुक्तं भवथा ब्रह्मन्! समस्तं सत्यमेव तत्।
मदृते मोहयित्रीह शङ्करस्य न विद्यते।।४२।।
Durga said, "O Brahma, whatever has been spoken by you, is utterly true. There is none else in the world to captivate Siva.
हरेऽगृहीतदारे तु सृष्टिर्नैषा सनातनी।
भविष्यतीति तत्सत्यं भवता प्रतिपादितम्।।४३।।
You have spoken a great truth that if Siva does not take a wife unto Himself, the creation cannot continue long.
ममापि मोहने यन्नो विद्यतेऽस्य महाप्रभोः।
त्वद्वाक्याद्द्वगुणो मेऽद्य प्रयत्नोऽभूत्स निर्भरः॥४४॥
I had also not been able to captivate the great lord, but at your Words my mind has been filled with double the enthusiasm.
अहं तथा यतिष्यामि यथा दारपरिग्रहम्।।
हरः करिष्यति विधे ! स्वयमेव विमोहितः॥४५ ।।
O Brahma, I shall do something by which - Siva getting captivated would himself accept a spouse.
सतीमूर्तिमहं धृत्वा तस्यैव वशवर्तिनी।
As Laksmi is the beloved of Visnu, similarly, I taking to the form of Sati shall obey the command of Siva.
After Marriage, Daksha didn't invite Shiva to Yajna and Sati immolated Herself and mingled with Adi Shakti or Durga .
Now all other Gods with Vishnu again wanted to make Durga Devi born in Saguna form for Rudra who was left without wife and became detached from wordly activities.
Chapter 3, Rudra Samhita (Parvati Khanda) says:
इति श्रुत्वा महीधरस्य वचनं ते सुरास्तदा।
ऊचुर्हर्यादयः प्रीताः सिद्धि मत्वा स्वकार्यतः।।१७।।
Brahma said, "Listening to the words of Himacala, Visnu and other gods, felt delighted, expecting the Success of their mission and they spoke.
हिमाचल! महाप्राज्ञ! शृण्वस्मद्वचनं हितम्।
यदर्थमागताः सर्वे तद्ब्रूमः प्रीतितो वयम्।।१८।।
The gods said, “O immensely wise Himacala, listen to our benevolent words. We shall now reveal the purpose for which we have arrived here.
या पुरा जगदम्बोमा दक्षकन्याऽभवद्रगिरे!।
रुद्रपत्री हि सा भूत्वा चिक्रीड सुचिरं भुवि।१९।।
Sati, the daughter of Daksa and the wife of Siva in earlier times, had been enjoying sports on earth in the form of Uma.
पितृतोऽनादरं प्राप्य संस्मृत्य स्वपणं सती।
जगाम स्वपदं त्यक्त्वा तच्छरीरं तदाम्बिका।२०।
Having been insulted by her father, remembering her pledge, getting rid of her human body, she achieved the supreme position.
सा कथा विदिता लोके तवापि हिमभूधरः। एवं सति महालाभो भवेद्देवगणस्य हि।२१।।
सर्वस्य भवतश्चापि स्युः सर्वे ते वशाः सुराः।।२२।।
The relevant story, O Himacala is well known to you. The gods too will be under your control.”
इत्याकण्र्य वचस्तेषां हर्यादीनां गिरीश्वरः।
तथास्त्विति प्रसन्नात्मा प्रोवाच न च अादरात्।।२३।।
Brahma said, "On hearing the words of Visnu and other gods, the delighted lord Himacala said, "Be it so.' Then he adored them with respect.
Thereafter Gods paryed to Durga Devi and Durga Devi agreed to born as daughter to couple of Menaka and Himachala.
Chapter 4 Rudra Samhita (Parvati Khanda) says
उमोवाच हे हरे! विधे! देवा! मुनयश्च गतव्यथा:।
सर्वे शृणुत मद्वाक्यं प्रसन्नाऽहं न संशयः।।२३।।
Uma said, "O Visnu, O Brahma, O other gods, let all of you be comfortable. Listen to my words. I am undoubtedly pleased with all of you.
चरितं मम सर्वत्र त्रैलोक्यस्य सुखावहम्॥
कृतं मयैव सकलं दक्षमोहादिकं च तत्।।२४।।
My performance always provides comforts to all the three worlds. The sport relating to the yajna of Daksa, was performed by me.
अवतारं करिष्यामि क्षितौ पूर्णं न संशयः।
बहवो हेतवोऽप्यत्र तद्वदामि महादरात्।।२५।।
I shall surely appear on earth, with full incarnation. There are several reasons for the same, which I am going to highlight with respect.
पुरा हिमाचलो देवा मेना चातिसुभक्तितः। सेवां मे चक्रतुस्तात! जननीवत्सतीतनोः।।२६।।
O Gods, earlier when I was born as Sati, then both Himalaya and Mena had served me lovingly as parents.
इदानीं कुरुतः सेवां सुभक्त्या मम नित्यशः।
मेना विशेषतस्तत्र सुतात्वे नात्र संशयः।।२७।।
Even presently both of them are serving me with utmost devotion. Out of both of them, Menaka, serves me particularly as a daughter.
Later Tarakasura performed Tapasya and got a boon from Brahma that only son born from Seed of Shiva can kill him. This made Gods to speed up their marriage related works.
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