What is the ruling of Sharia on woman shaking hands with non-Mahram man?
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) said: "It is better to be stabbed in the head than touch a woman to whom one isn`t related." [Transmitted by Al-Tabarani /Al-Mo`jam Al-Kabir, Hadith No.16880]. Accordingly, it is forbidden for men to shake hands with non-Mahram* woman. And Allah The Almighty Knows Best.
* The women/men that you are allowed to marry. It includes all women/men other than mahram (including cousins).
My headcloth is causing me a terrible headache and severe shortsightedness. Is it permissible for me to take it off?
All perfect praise be to Allah, the Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad, sallallahu ‘alayhi wa sallam, is His slave and Messenger.
If the headcloth Khimar is harmful, it is sufficient to wear the Niqab (face-veil). However, with the passage of time you will get used to the Khimar and these symptoms will go away. And Allah The Almighty Knows Best.
Is a woman`s prayer considered invalid if non-Mahrams (marriageable men) saw her offering it?
A woman`s prayer isn`t invalidated if non-Mahrams saw her offering it, but she had better pray in isolation.
Is it permissible for the man to go out wearing shorts?
All Perfect Praise is due to Allah, The Lord of The Worlds
It is permissible so long as his Awrah (private part), which is between his navel and knee, is covered. However, the regular wear traditions and modesty are to be observed. And Allah Knows Best