Is it permissible to make up for missed fast after the beginning of the second half of the month of Sha`ban(the month before Ramadhaan) ?

Yes, it is permissible, but one who had missed fasts should hasten to make up for them. As for the Hadith mentioned in this regard, the prohibition is for offering absolute voluntary fasting.

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 is usually said upon the slaughtering of the Aqeeqah (the sheep slaughtered on the seventh day from the child`s birth)?

It is desirable to say: “Bismillah Alrahman Alraheem, was`salatu Wa s`salamu Ala Sa`idenah Mohammad, O Allah, this Aqeeqah is from You and to You on behalf of …..”

Is it permissible for one to shake hands with his uncle`s daughter?

No, it isn`t because he is a non-Mahram(Marriageable) to her.