Is it permissible to recite the Holy Qur'an or to play a recording at mourning gatherings? Or is it considered an innovation?
Whenever Qur'anic verses are recited, it is obligatory upon everyone to listen, otherwise; they should remain silent. And Allah Knows Best.
Is the son of the sister by breastfeeding considered a mahram in Islam, and is it permissible for a woman to go for Hajj with him?
Everything that is prohibited by lineage is also prohibited by breastfeeding. The son of the sister by breastfeeding is considered a mahram, just like the son of the sister by blood, and it is permissible for him to be her mahram for Hajj and Umrah. 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.