In the sunnah we find this hadith in sunan ibn Majah (Qualified as da'if):
“The Messenger of Allah (ﷺ) said: ‘Whoever vomits, has a nosebleed, belches, or emits prostatic fluid, should stop praying; perform ablution, then resume his prayer, and while he is in that state he should not speak.”
this seems to be taken literally by the hanafi school of fiqh (Who says that one can continue a prayer after interrupting it for legal reasons?).
In his Muwatta' imam Malik quotes the following narrations:
Yahya related to me from Malik from Nafi that Abdullah ibn Umar would leave and do wudu if he had a nose-bleed and then return and complete his prayer without saying anything. (al-Muwatta')
Yahya related to me from Malik from Nafi that Abdullah ibn Abbas used to have nose-bleeds and would leave to wash off the blood. He would then return and complete his prayer. . (al-Muwatta')
which seem to agree with the above hadith while you further find the following narrations, saying otherwise:
Yahya related to me from Malik from Yazid ibn Abdullah Qusayt al- Laythi that he saw Said ibn al-Musayyab having a nose-bleed while praying. He went off to the room of Umm Salama, the wife of the Prophet, may Allah bless him and grant him peace, and water was brought to him and he did wudu. He then returned and completed his prayer. (al-Muwatta')
Yahya related to me that Abd ar-Rahman ibn Harmala al-Aslami said, "I saw Said ibn al-Musayyab with his nose bleeding and blood poured out of it so that his fingers were all red from the blood coming out of his nose, and he prayed without doing wudu." (al-Muwatta')
Yahya related to me from Malik from Abd ar Rahman ibn al-Mujabbar that he saw Salim ibn Abdullah with blood running from his nose so that his fingers were all coloured red. Then he rubbed it and prayed without doing wudu. (al-Muwatta')
which lead to the conclusion that it is not necessary to perform wudu' if one has nose bleeding.
I'd like to know what are the views of scholars or madhhabs on the matter of vomiting and nose bleeding (if a focus is necessary due to the complexity of the question let's restrict it on nose bleeding).