Is it permissible for us to sever ties of kinship if our blood-relatives` gatherings/meetings involve acts of sin?

Observing kinship ties is obligatory, and paying your blood-relatives a visit causing their sinful acts to be hindered , then you should do so. However, if their sinful acts continue while you are at their gathering and they didn't respond positively, then observing kinship ties via telephone and the like will suffice in this case. And Allah Knows 

Does the clipping of nails nullify ablution?

No, it does not, and it is preferable to wash the hands afterwards.

What is the ruling on reciting the Chapter after Al-Faatihah , and what should a worshiper who forgets it do?

All perfect praise be to Allah, The Lord of The Worlds.                                                                                                                                                                  Any other part of Qur'an that a worshiper reads after Al-Faatihah during the first and the second Rak`ahs is a Sunnah, and forgetting to do so doesn`t nullify prayer, and doesn`t require performing Sujood As-Sahw (prostration of forgetfulness). And Allah Knows Best.

Is it permissible to go to the bathroom wearing a necklace that has the word "Allah" inscribed on it?

All Perfect Praie be to Allah, The Lord of The Worlds, and may His Peace and Blessings be upon our Prophet Muhammad and upon all of his family and companions.

Anything that contains words pertaining to Allah, The Almighty, is desired to be left outside when going to the bathroom, or to be hidden for fear of losing it. And Allah Knows Best.