• Is a woman`s nephew by suckling considered a Mahram( i.e. unmarriageable)who is permitted to accompany her to Hajj?
All that is prohibited by lineage is prohibited by suckling, and a nephew by suckling is a Mahram like a nephew by lineage ;therefore, it is permissible for him to be a Mahram for his aunt in Hajj and Umrah.
• A man who was on travel prayed Duhr as four Rakhas upon leaving Tafilah heading to Amman. However, on his way to Amman, he prayed Asir as two Rakhas (Shortened). Is what he did correct from an Islamic perspective?
All perfect praise be to Allah, The Lord of The Worlds. May His blessings and peace be upon our Prophet Mohammad and upon all his family and companions.
What he did is correct; he is liable for nothing before Allah the Almighty since combining and shortening prayers during travel are two separate concessions. Therefore, it is permissible for a traveler to shorten prayer without combining it with another. And Allah the Almighty knows best.
• Is it permissible for a menopausal woman who has blood discharges to fast ?
If she reached the age of menopause, and her menses stopped, but later she saw blood discharges that lasted a complete day and night (24) hours, then it is menstrual blood. Still, if these blood discharges lasted less than a complete day and night (24) hours, then it is not menstrual blood, but it is bleeding outside the monthly period(Istihadah), and it does not prevent her from prayer and fast, but she has to make ablution for every prayer.
• 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.