Which hour coincides with the last third of the night?

This hour could be figured out by calculating the number of night hours which start at sunset, and end at dawn break, then the total is divided on three to get the result.

Is it permissible for a young man and a young woman to enter into a covenant as a married couple without having a true seclusion (Khalwah) until their marriage contract is officially concluded?

The relation which isn`t based on a marriage contract is forbidden, and entering into a covenant as a married couple is unaccounted for and is from the evil suggestions of the devil.

What is the expiation for perjury?

One who commits perjury should repent, seek Allah`s forgiveness and offer an expiation which is: feeding ten indigent persons, or clothing them, or giving a slave his freedom, but if that is beyond his means then, he should fast for three days. Allah, The Almighty, Says in this regard (What means): "Allah will not call you to account for what is futile in your oaths, but He will call you to account for your deliberate oaths: for expiation, feed ten indigent persons, on a scale of the average for the food of your families; or clothe them; or give a slave his freedom. If that is beyond your means, fast for three days. That is the expiation for the oaths ye have sworn. But keep to your oaths. Thus doth Allah make clear to you His signs, that ye may be grateful." [Al-Ma`idah/89].

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