I used artificial nails on my wedding night, but I was surprised to find out that they require at least a week to remove. However, I performed all my prayers during that period. What is the ruling on those prayers?

 

If these nails prevent water from reaching one's skin then your prayer are invalid and obliged to redo all of them. And Allah Knows Best.

Should one who doesn`t perform prayer out of laziness make it up later, and how should he/she do so?

All perfect praise be to Allah,The Lord of The Worlds                                                                                                                                                              He/she is obliged to make up missed prayers by offering with each obligatory prayer another one, and if he/she offers two, then it is better. And Allah Knows Best.

Is it obligatory for a wife who has been abandoned for many years to observe the waiting period (iddah) after her husband passes away?

If her husband passes away, the wife is required to observe the waiting period (Iddah), even if he had abandoned her, because she is his wife and inherits from him, so she must observe the iddah after his death. And Allah Knows Best.

Is wearing a cap permissible for boys, or girls?

All Perfect Praise is due to Allah, The Lord of The Worlds.

In principle, anything is permissible sol long as there is no Sharia evidence prohibiting it, thus wearing the cap by boys is permissible as long as it isn`t done in imitation of the disbelievers. Moreover, a Muslim woman should avoid all that which attracts the attention of men because whatever leads to the forbidden is forbidden. And Allah Knows Best.