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.

Is it permissible for the woman to use her mobile phone while on the street?

All Perfect Praise is due to Allah, The Lord of The Worlds 

Modesty should be observed by the woman at all times, so she is free to use her phone while on the street as long as it is done within the limits of Islamic manners. And Allah The Almigty Knows Best.

During a flight from Amman to the UAE, while we were flying over Saudi Arabia, the captain announced that it was time to break our fast, coinciding with the exact time of sunset. After I had broken my fast with my first date, the captain apologized for the earlier announcement and clarified that the correct time would be in 20 minutes. Am I required to continue fasting on that day?

Whosoever break his fast at teh very first announcement to make up the missed day of Ramadan, since Allah, The Most Exalted, Said (What means): "then complete your fast Till the night appears" [Al-Baqara/187]. And Allah Knows Best.

What is the ruling on someone who prays Fajr late, after sunrise, knowing that they wake up on time but return to sleep and do not perform it within its time?