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.

What is usually said upon the slaughtering of the Aqeeqah (the sheep slaughtered on the seventh day from the child`s birth)?

It is desirable to say: "Bismillah Alrahman Alraheem, was`salatu Wa s`salamu Ala Sa`idenah Mohammad, O Allah, this Aqeeqah is from You and to You on behalf of ….."

Is it permissible to fast on behalf of the dead who died owing missed fast to make up?

His relative should fast on his behalf, and it is permissible for the guardians to give permission to non-relatives of the dead to fast on his behalf as well. And Allah Knows Best.

Is prayer nullified by having thoughts while performing it?

Prayer isn`t nullified by having thoughts while performing it, provided that the praying person offers all of its pillars and meets all of its conditions. However, the reward decreases significantly in the absence of reverence since a praying person is rewarded according to the attentiveness of his/her heart during prayer.