What should one who made an intention, at night, to make up for a missed fasting day, but broke his fast on the next day?
One who started making up a missed fasting day, then broke his/her fast without a legitimate reason is considered sinful, and is only obliged to make up for the missed fasting days. And Allah Knows Best.
What is the ruling on a Muslim who slaughters an animal while being in a state of Janabah (ritual impurity)?
All Perfect Praise be to Allah, The Lord of The Worlds, and may His Peace and Blessings be upon our Prophet Muhammad and upo all of his family and companions.
Slaughtering is permissible for the ritually impure man and the menstruating woman because ritual purity isn`t a condition for the validity of the slaughtering. Moreover, it is permissible for the uncircumcised person to do the slaughtering as well. And Allah The Almighy Knows Best.
How is night prayer offered?
Night prayer is offering voluntary prayer after Maghrib and before Fajr (Dawn). As for Tahajjud, it is offering voluntary prayer at night after waking up voluntarily, and for Allah`s sake as He Says (What means): "And pray in the small watches of the morning: (it would be) an additional prayer (or spiritual profit) for thee: soon will thy Lord raise thee to a Station of Praise and Glory!" [Al-Isra`/79]. Offering Tahajjud is better than offering voluntary prayer before going to bed.
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