Q 1: What is the ruling on a male student shaking hands with his female colleague? What should he do if she extends her hand to shake his?
A: It is not permissible to study in mixed institutes where males and females occupy the same facility, school, or seats. This is one of the gravest ways leading to Fitnah (temptation). Thus, it is not permissible for a male or female student to place themselves in this situation leading to Fitnah.
Moreover, it is not permissible for a Muslim to shake hands with an Ajnabiyyah (woman lawful for the man to marry), even if she extends her hand to him. In this case, the Muslim should inform her of the impermissibility of shaking hands with her. It is confirmed that the Prophet (peace be upon him) said upon taking the pledge of allegiance from women, "I do not shake hands with women."
It is also related on the authority of `Aishah (may Allah be pleased with her) that she said, "By Allah! The hand of Allah's Messenger (peace be upon him) never touched the hand of any woman, but he only used to take their pledge of allegiance orally."
Indeed in the Messenger of Allâh (Muhammad ﺻﻠﻰ ﺍﷲ ﻋﻠﻴﻪ ﻭﺳﻠﻢ) you have a good example to follow for him who hopes for (the Meeting with) Allâh and the Last Day, and remembers Allâh much.
However, there is no harm in extending Salam (Islamic greeting of peace) that does not involve any Fitnah, shaking hands, evil intentions, seductive speech, or Khulwah (being alone with a member of the opposite sex), and while a woman is veiled.
O wives of the Prophet! You are not like any other women. If you keep your duty (to Allâh), then be not soft in speech, lest he in whose heart is a disease (of hypocrisy, or evil desire for adultery) should be moved with desire, but speak in an honourable manner.
As for women shaking hands with other women or with Mahram men (spouse or unmarriageable relative) such as the father, brother, uncle and others, there is no harm in this. Allah is the Grantor of success.