Is it permissible for us to sever ties of kinship if our blood-relatives` gatherings involve acts of sin?

Maintaining kinship ties is mandatory, thus if visiting your blood-relatives hinders their sinful acts , then you should do so. However, if their sinful acts persist while you are at their gathering, then it is sufficient that you maintain kinship ties via telephone and the like. And Allah Knows 

Is the `Iddah (waiting period) of the divorced effective from the time the divorce took place, or from the time of registering it at the court?

It is considered effective from the time the divorce took place, and not from the time of registering it at the court. And Allah Knows Best.

What should a sick person, who is incapable of fasting, and who didn`t make up for missed fasting days, do?

One who has missed fasting days is obliged to make up for them, but if he/she wasn`t able to because of an incurable disease , or old age, then he/she has to pay a ransom which is feeding a needy person for each of the missed fasting days. 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.