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.

What is the ruling on giving my mother a gift at "Mother's day", is it Halal or Haram?

Praise be to Allah the Lord of the Worlds.

There is nothing wrong with doing that as long as it isn`t done in imitation of the disbelievers. And Allah The Almighty Knows Best.

Is a wife considered sinful if she asked for divorce because her husband married another woman?

A second marriage is lawful, and so it is impermissible for you to ask for divorce because of that. If you are certain that you won`t be able to deliver Allah`s rights and those of your husband`s , then it is permissible for you to ask for Khul`a (Divorce on the demand of the wife). However, you should exercise patience since after difficulties, there is easiness by the will of Allah.

Does the wife have the right to stop her husband from visiting some of his relatives because their meetings involve religious violations?

Couples should advise one another in kindness, and the wife can stop her husband from committing sin by resorting to a kind and tactful approach.