What should a person who doubted having missed a pillar of prayer do?

All praise be to Allah, The Lord of The Worlds.                                                                                                                                                                              The prayer of one who doubts having missed one of its pillars is valid, and nothing is due on him/her since having a doubt after offering the act of worship doesn`t undermine its validity. 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.

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 it permissible for a Muslim to escort his deceased disbelieving relative to his final resting place?

It is permissible provided that he doesn`t walk behind non-Islamic symbols, rather he should walk in front of them.