Is it permissible for men to wear silver bracelets and necklaces?

All Praise is due Allah, The Lord of The Worlds

Men are allowed to wear silver rings, but they aren`t allowed to wear bracelets and necklaces because by doing so they imitate women. And Allah Knows Best.

What should one who forgot to offer the second Sajda during congregational prayer, and remembered it before the end of the prayer do?

All perfect praise be to Allah, The Lord of The Worlds.                                                                                                                                                          Whoever forgets a pillar of prayer while in congregation must follow the lead of the Imam, and after the latter offers Tasleem, he should offer one Rakah, then perform Sujood As-Sahw (prostration of forgetfulness). And Allah Knows Best.

Does an internal examination require ghusl?

Ghusl isn't required for making internal examination; however, it invalidates one's fasting. And Allah Knows Best.

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.