Is touching a non-Mahram woman invalidates person's ablution?

Touching non-Mahram woman is prohibited, and according to the Shafiee's, it invalidates the ablution. And Allah Knows Best.

What is permissible for the husband to do as far as having sex with his wife is concerned?

The couple are free to enjoy each other sexually provided that they don`t do it during menses. It is also forbidden for the husband to enjoy the wife from her anus.

Does the clipping of nails nullify ablution?

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

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.