Is it permissible to make ablution without answering the call of nature after waking up?
Washing front and back openings (Istinjaa`) is not a condition for the validity of ablution because it is done to remove Najaasah (impurity), thus if there is no Najaasah, there is no need for that, and then it is permissible to make ablution without answering the call of nature. However, prayer is invalidated if there is Najaasah on the anus ,or the front opening, so it (Najaasah) must be removed for the prayer to become valid, and not the ablution.
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.
Does the clipping of nails nullify ablution?
No, it does not, and it is preferable to wash the hands afterwards.
Is it permissible for a person who had vowed to give a certain amount of money to another, but didn`t find the latter to donate that money to the mosque?
In principle, the vowing person should abide by his vow as much as possible. Therefore, if he couldn`t find the person that he had made the vow for, then the vow itself is countless and nothing is due on its maker. However, if the latter happens to find the former later on then, he has to give him that money.