Nikah without the guardian or wali

a) The Madhab of Imam Malik and ash-Shafiee.
b) The Madhab of Dhaahiriyah (i.e. Dawud ad-Dhaahiri)
c) The Madhab of Abu Hanifa, Zafar, Sho`abi and az-Zuhri
d) Hanbali Madhab.
e) Who will be the Wali of The woman who has no guardian and has no approach to reach the Muslim Leader or the Qaadi.

a) The Madhab of Imam Malik and ash-Shafiee.

Without the wali nikah is invalid.

Evidence:

The Prophet (peace and blessings of Allaah be upon him): “There is no marriage without a wali.” Narrated by Abu Dawood (2085), al-Tirmidhi (1101) and Ibn Majaah (1881) from the hadeeth of Abu Moosa al-Ash’ari; classed as saheeh by al-Albaani in Saheeh al-Tirmidhi.

Along with other Qur`anic evidences.

b) The Madhab of Dhaahiriyah (i.e. Dawud ad-Dhaahiri)

If the girl is virgin, the nikah is invalid without wali and if she has been previously married then there is no need to take consent of wali.

Evidence:

A woman who has been previously married (Thayyib) HAS MORE RIGHT TO HER PERSON THAN HER WALI; and a virgin’s father must ask her consent from her, her consent being her silence, At times he said: Her silence is her affirmation.
Sahih Muslim 1421

Along with other evidences as well.

c) The Madhab of Abu Hanifa, Zafar, Sho`abi and az-Zuhri

If the woman (virgin or who has been previously married), marries a man compatible to her without her wali, this allowed.

Evidence:

“..but if they (wives) leave, there is no sin on you for that which they do of themselves..” [2:240] [above is precised from the book Bidayat al Mujtahid by Ibn Rushd 3/36]

d) Hanbali Madhab.

Same as Maaliki and Shafiee Madhab. Add the following,

Shaykh al Islam Ibn Taymiyah said:
وإذا رضيت رجلا وكان كفؤا لها وجب على وليها كالأخ ثم العم أن يزوجها به فإن عضلها وامتنع من تزويجها زوجها الولي الأبعد منه أو الحاكم بغير إذنه باتفاق العلماء فليس للولي أن يجبرها على نكاح من لا ترضاه ولا يعضلها عن نكاح من ترضاه إذا كان كفؤا باتفاق الأئمة وإنما يجبرها ويعضلها أهل الجاهلية والظلمة الذين يزوجون نساءهم لمن يختارونه لغرض لا لمصلحة المرأة ويكرهونها على ذلك أو يخجلونها حتى تفعل ويعضلونها عن نكاح من يكون كفؤا لها لعداوة
أو غرض وهذا كله من عمل الجاهلية والظلم والعدوان وهو مما حرمه الله ورسوله واتفق المسلمون على تحريمه وأوجب الله على أولياء النساء أن ينظروا في مصلحة المرأة لا في أهواءهم كسائر الأولياء والوكلاء ممن تصرف لغيره فإنه يقصد مصلحة من تصرف له لا يقصد هواه فإن هذا من الأمانة التي أمر الله أن تؤدي إلى أهلها فقال إن الله يأمركم أن تؤدوا الأمانات إلى أهلها وإذا حكمتم بين الناس أن تحكموا بالعدل وهذا من النصيحة الواجبة وقد قال النبي الدين النصيحة الدين النصيحة الدين النصيحة قالوا لمن يا رسول الله قال لله ولكتابه ولرسوله ولأئمة المسلمين وعامتهم والله أعلم
And when she is willing to marry a man and he is also suitable for her, then it is obligation on her guardian that he should marry her with him, and if he (the guardian) stops her from getting married or refrain from it, then according to the agreement of the scholars the next closest relative or the Haakim will make her nikah without the permission of first guardian. Those who prevent her (from getting married where she is willing) and forcing her (to marry other person) are on the way of oppressors at the time of ignorance, who in their guardianship wants the women to marry those people who are not accepted by women rather they are accepted by themselves. Then they force her and make her ashamed until she marries a person of their choice, and they stop her from marrying a suitable person for her due to their enmity or any other purpose. All of this is the practice of the days of al Jaahiliyah and oppression and enmity, which is prohibited according to Allah and his Messenger (peace be upon him), and It is prohibited according to the consensus of the scholars. Allah obligates the guardians of women to see the interest of women rather than following their desires… This is the same amanah which Allah mentioned that it is obligatory to fulfill.
“Indeed, Allah commands you to render trusts to whom they are due and when you judge between people to judge with justice.” [4:58] This well wishing (Sincerity) is obligation as the Messenger of Allah said:
“Religion is sincerity, religion is sincerity (Al-Nasihah), religion is sincerity.” They said; “To whom, O Messenger of Allah?” He said: “To Allah, to His Book, to His Messenger, to the imams of the Muslims and to their common folk.” [Sunan an-Nasa’i 4199] and Allah knows best.[Majmu al Fatawa (32/52-53)]

e) Who will be the Wali of The woman who has no guardian and has no approach to reach the Muslim Leader or the Qaadi.

al Qurtubi said:
وإذا كانت المرأة بموضع لا سلطان فيه ولا ولي لها فإنها تصير أمرها إلى من يوثق به من جيرانها ، فيزوجها ويكون هو وليها في هذه الحال ؛ لأن الناس لابد لهم من التزويج ، وإنما يعملون فيه بأحسن ما يمكن ، وعلى هذا قال مالك في المرأة الضعيفة الحال : إنه يزوجها من تسند أمرها إليه ؛ لأنها ممن تضعف عن السلطان فأشبهت من لا سلطان بحضرتها ، فرجعت في الجملة إلى أن المسلمين أولياؤها
If the woman is at place where there is no Muslim leader or her guardian, Then she can give her case to her trustworthy (Muslim) neighbor who will be her guardian in marriage. Because marriage is must for people, so they should do it in a best way whatever is possible. Imam Malik said regarding the woman who is in weak condition (i.e. who has no wali and can not reach to sultan). She can give her case to (the trusted) person who can be her guardian in her marriage, because she can not reach the leader, her case is like the leader is not present (and she wants to marry). So, in conclusion Muslims are her guardians. [Tafseer Ahkam al Quran, Surah al Baqarah verse 221]