What is the ruling on working for a company that manufactures cigarettes?

All Perfect Praise is due to Allah, The Lord of The Worlds

Working for a company that manufactures cigarettes is forbidden because smoking itself is forbidden, so its production, selling and circulation are forbidden as a result since they aid in committing sin, and whoever aids in that is like the sinner himself. And Allah Knows Best.

Is it permissible for a menstruating woman to recite the Quran from the computer without actually touching the Quran?

It is not permissible for a woman in her menstrual period, or in postpartum to recite the Quran, even by heart, or without actually touching the Quran itself. Recitation itself is what is prohibited on her regardless of how it is done. However, it is permissible for her to surf through the Quran by her eyes, or to recall it in her head without uttering the words. There is no harm for her to look at the Quranic verses on the computer without touching it, or to utter the words as this is called looking not reciting or reading.

What should a woman, who has given several births during different months of Ramadhaan, and didn`t make up for them in addition to forgetting the exact number of the days and years in which she had missed fasting, do?

She should make up the missed days of Ramadan after estimating their number, and paying the ransom (in food) due on each missed day that she had delayed making up. In addition, she is obliged to repay the ransom according to the number of years if she was able to fast before that time, but didn`t.

What is the best manner in distributing the meat of the Aqeeqah?

It is better to divide all of it as cooked meat amongst the poor and the needy. Dividing it into three portions is a Sunnah, but it is permissible to keep all of it provided that a portion, even a small one, is given to the poor and needy (one kilo for example).