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 is the ruling on wiping the head during ablution from behind a barrier like a headscarf (hijab)?
The obligation in ablution is to wipe some of the head; it is not a condition to wipe all of it. As for one who has a turban or headscarf on their head, it is permissible for them to wipe over it after wiping a part of their head in any place. This is easily done at the front of the head. This means the woman begins by wiping the front of her head from under the headscarf, then completes the wiping to the back of the head from over the headscarf. If the woman wipes over the headscarf on her head and the moisture reaches the scalp or some hair within the boundaries of the head, it suffices for wiping the head. And Allah the Almighty knows best.
Is it permissible for the young woman to turn the suitor down if she had no feelings for him?
Yes, it is permissible for her to do so.
What is the ruling on a person in a state of major impurity (junub) walking or eating before performing the ritual bath?
It is from the Sunnah for a Muslim to hasten to perform the ritual bath for major impurity. However, if he delays it, he does not sin provided he does not miss the prayer. It is permissible for the junub to walk, eat, drink, and sleep. It is recommended for him to perform ablution before these actions to reduce the state of impurity, although the best is for him to perform the ritual bath. And Allah the Almighty knows best.