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

I work for a company whose manager refuses to allow employees to leave during work hours. Do I incur sin if I secretly went out to join the congregation at the Masjid? Is doing so considered a breach of trust?

Praise be to the Lord of the Worlds.

We recommend that you pray at work to encourage your colleagues to follow suit, because whoever guides to good has the like of the reward of the person who actually does it. And Allah The Almighty Knows Best.

What are the conditions of the Aqeeqah?

The animals that are acceptable to be slaughtered for 'Aqeeqah are the same which are acceptable to be slaughtered for the sacrifice of 'Eed in terms of their age and lack of defects. The sheep should be one year old, and its teeth should have fallen off. As for goats and cows, they should have finished two years of age and entered into the third. The Aqeeqah should be free of any defects as regards meat, or price value.

Is it permissible to combine and shorten prayers at home before traveling?

It is impermissible to combine and shorten prayers at home before traveling because a person is called a traveler once he/she departs the borders of his/her country, and offering prayer that way was originally intended to exempt a traveler. And Allah Knows Best.