How are gaps filled during congregational prayer?

A praying person is allowed to take two steps in order to fill the gap in the row ahead. If the distance was long and requires much movement, then it is impermissible for one to move to fill it because much consecutive movement (three movements) during prayer invalidates it.

What should a woman who delayed making up for missed fast, due to menstruation, till the start of the next Ramadan?

Whosoever broke fasting in Ramadan, and was able to make up for it, but didn`t until the next Ramadan started, is obliged to make up for the missed days, and to pay the ransom as well. However, if he/she wasn`t able to before next Ramadhaan due to an excuse, then he/she should only make up for the missed days. And Allah Knows Best.

What is the difference between Tahajjud prayer and night prayer, and do they have a particular Witr prayer other than that of Isha (evening prayer)?

Tahajjud and Qiyam-Al-Lail (night prayer) are two words for the same meaning which is offering voluntary prayer at night after sunset, but before dawn break. However, Tahajjud is offered after waking up; whereas, Qiyam-Al-Lail is permissible before sleeping, or after it. Therefore, every Tahajjud is a Qiyam, but not every Qiyam is a Tahajjud. Moreover, there is no specified Witr for both of them, but Witr after Isha is by itself the Witr of Qiyam, and it is a Sunnah to postpone it until after Qiyam if the worshiper thought that he was most probably going to wake up in order to offer it, but if he wasn`t sure of waking up, then he is at liberty to offer it before going to bed as was reported in the Prophet`s Hadith.

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