Is prayer nullified by nose bleed?

 All praise be to Allah, The Lord of The Worlds.                                                                                                                                                                           Blood coming out of the nose, or a wound doesn`t nullify prayer regardless of its quantity. And Allah Knows Best.

Does post-natal bleeding (Nifas) stop before forty days after childbirth?

What is the ruling on performing dry ablution (tayammum) instead of the ritual bath due to severe cold?

Whoever is sick such that he cannot use water, or finds the water cold and heating it is not possible, and the person would be harmed by using cold water, it is permissible for him to perform tayammum. He must make up what he prayed with tayammum, because this is a place where there is water. And Allah the Almighty knows best.

What is the ruling on the cessation of blood after (40) days from delivery, but later continued sporadically during two days of Ramadan?

Once postpartum bleeding (Nifas) ceases, and the woman is certain that it won`t reoccur, then she becomes ritually pure and so she is free to make Ghusl (purificatory bath), pray, and fast. If the bleeding reoccurs before fifteen days from its cessation, and before the end of (60) days after delivery, then the ruling on postpartum bleeding is effective, and her fasting and prayer are null and void, thus she must make up the fasting that she missed and not the prayer during those particular days.