Is it permissible to swear by (Allah "The Living")?
Yes, it is permissible because "The Living" is one of Allah`s Attributes.
A man who was on travel prayed Duhr as four Rakhas upon leaving Tafilah heading to Amman. However, on his way to Amman, he prayed Asir as two Rakhas (Shortened). Is what he did correct from an Islamic perspective?
All perfect praise be to Allah, The Lord of The Worlds. May His blessings and peace be upon our Prophet Mohammad and upon all his family and companions.
What he did is correct; he is liable for nothing before Allah The Almighty, since combining and shortening prayers during travel are two separate concessions. Therefore, it is permissible for a traveler to shorten prayer without combining it with another. And Allah The Almighty Knows Best.
If the imam does not leave time for reciting Surah Al-Fatihah, is the recitation of Al-Fatihah waived for the follower?
The recitation of Al-Fatihah is not waived for the follower. They should recite it silently, even if it causes a slight delay in following the imam in bowing (Ruku'). And Allah Knows Best.
What is the ruling on going out wearing a long loose skirt with a long shirt that are neither tight nor revealing and without wearing any makeup?
All perfect praise be to Allah the Lord of the Worlds. May Allah`s peace and blessings be upon our Prophet Mohammad and upon all his family and companions.
This is permissible as long as it doesn`t attract the attention and isn`t considered adornment, and this varies according to the environment in which you live. And Allah The Almighty Knows Best.