Witr is done after 'Isha salah. There is no doubt about that and no difference in opinion between different schools because the Prophet clearly said:
“Allaah has prescribed for you a prayer (by which He may increase your reward), which is Witr; Allaah has enjoined it for you during the time between ‘Isha’ prayer until dawn begins.” Narrated by al-Tirmidhi, 425; classed as saheeh by al-Albaani in Saheeh al-Tirmidhi.
It can be done directly after 'Isha or in the later parts of the night.
“Whoever fears that he will not get up at the end of the night, let him pray Witr at the beginning of the night, but whoever thinks that he will be able to get up at the end of the night, let him pray Witr at the end of the night, for prayer at the end of the night is witnessed (by the angels) and that is better.” Narrated by Muslim, 755.
The Hanafis and Shafi'is have difference in how Witr is prayer, whether it is three rak'ah together or two then one. Both are mentioned in the sunnah, so doing either is fine. One isn't "better" or "more correct" than the other like how some people say.
“He used to pray Witr with three rak'ahs and he did not sit except in the last of them.” Narrated by al-Nasaa’i, 3/234; al-Bayhaqi, 3/31. al-Nawawi said in al-Majmoo’ (4/7): it was narrated by al-Nasaa’i with a hasan isnaad, and by al-Bayhaqi with a saheeh isnaad.
Saying the tasleem after two rak'ahs, then praying one rak’ah on its own, because of the report narrated from Ibn ‘Umar (may Allaah be pleased with him), that he used to separate the two rak'ahs from the single rak'ah with a tasleem, and he said that the Prophet (peace and blessings of Allaah be upon him) used to do that. Narrated by Ibn Hibbaan (2435); Ibn Hajar said in al-Fath (2/482): its isnaad is qawiy (strong).
The du'a can be read before or after bowing; both are mentioned in the sunnah. But reciting it after bowing is mentioned more in the sunnah. But this does not make it more righteous than the other.
Shafi'is usually recite "Allahum mahdini fii man hadayt..." while Hanafis recited "Allahuma innaa nasta iinuka..." but both are mentioned in the sunnah and are correct in their own ways.
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