What is the ruling on having brotherly ties between a strange man and a strange woman, and is the former considered a Mahram (unmarriageable) to the latter?

Such a relation between a strange man and a strange woman is forbidden in Islam because it involves forbidding what Allah has allowed by marriage, and allowing what Allah has forbidden such as looking and the like. Such a relation doesn`t render any act lawful between them, thus they are forbidden to look at each other, have a Khulwah (seclusion), and travel together.

Is it permissible for a person to give the Zakah (obligatory charity) to his daughter-in-law?

Yes, it is permissible for a person to give the Zakah to his daughter-in-law if she was poor, and none provided for particularly by whom are obliged to provide for her. And Allah Knows Best.

What is the ruling on taking part in an online contest to win a prize of 100,000 dollars in return for answering thirty questions in different fields of knowledge and science? Knowing that there is no drawing of lots; only one or two cards are purchased to commence the contest.

Is it permissible to move a grave from its current location to another place, for example, from a garden in a home to a cemetery, knowing that the grave has been there for a long time, about fifty years?

It is not permissible to move the deceased from his grave, as this may result in breaking his bones and violating his sanctity. Such actions are not allowed. And Allah Knows Best.