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 Ramadan)?
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. And Allah Knows Best.
Is permissible to buy/sell with non-Muslim?
Yes buying/selling for a Muslim with non-Muslim is permissible as regards what is lawful.
Is a woman`s voice 'Awrah even when praising the Prophet (PBUH) via satellite channels?
Praise be to Allah the Lord of the Worlds. May His peace and blessings be upon our Prophet Mohammad and upon all his family and companions.
The Sharia term 'Awrah has two meanings:-
First: `Awrah that invalidates prayer when uncovered, and in this sense a woman`s voice isn`t considered `awrah. Second: `Awrah that causes men to be tempted by women and vice versa. Moreover, if a woman`s voice is soft or alluring, then it is forbidden for her to speak deliberately before strange men. This is because Almighty Allah Says (What means): "O ye wives of the Prophet! Ye are not like any other women. If ye keep your duty (to Allah), then be not soft of speech, lest he in whose heart is a disease aspire (to you), but utter customary speech." [Al-Ahzab/32]. However, a man should avoid listening whatever incites his desires. And Allah The Almighty Knows Best.
Is it permissible to shorten the eyebrows if they get too long?
Yes, it is permissible, but not desirable.