What is the ruling when the duration of the monthly period varies, and sporadically increases?

If the period didn`t exceed (15) days from the first seeing of blood discharges until its end including the times of temporary cessation, then it is menses. But, if the period exceeded (15) days, then the woman experiencing this is Mostahadah (bleeding outside the time of monthly period), and should abide by her regular period, and consult one of the scholars.

Should I pick up the newspapers thrown at the street because they definitely contain sacred names?

All perfect praise be to Allah the Lord of the Worlds. May His peace and blessings be upon our Prophet Mohammad and upon all his family and companions.

The Messenger of Allah (PBUH) says: "Be mindful of Allah and He will protect you." Part of this is protecting whatever contains the name of Allah. Therefore, if you see a paper containing Allah's name thrown in the street, then you should grab it and similar papers, then burn them in a clean place. Doing so is permissible because Othman (May Allah be pleased with him) has burnt copies of the Quran which didn`t match the main Mus-haf. However, you aren`t required to collect every paper in the street. And Allah The Almighty Knows Best.

Does touching one's wife invalidate ablution?

Touching the wife according to the Shafi'ee's doctrine invalidate husband's ablution. And Allah Knows Best.

What is the expiation for breaking fast due to being on a journey, or being sick, or being in a state of menstruation?

No expiation is due on the aforesaid categories, but they are obliged to make up for the missed fast. However, if any of them failed to do so while being able to, and the next Ramadhaan has come, then making up for those days is incumbent on him/her , and paying the ransom as well. And Allah Knows Best.