Is it permissible to shorten the eyebrows if they get too long?
Yes, it is permissible, but not desirable.
Should one obey his mother even if she was wrong?
Obeying one`s mother is mandatory in matters that are beneficial to her and as long as she drive her children to commit sin.
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 incumbent on a wife whose husband hasn`t gone to bed with her(for sexual intercourse) for many years to observe `Iddah(waiting period) upon his death?
Upon the death of the husband, a wife is obliged to observe the `Iddah even if he hadn`t slept with her because she inherits him.