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.

A pious man proposed to me and my father was hesitant in this regard because he is black, am I sinful if I accepted his proposal?

If the suitor is pious, color isn`t a drawback. However, try talking to your father kindly, so that he approves of your marriage with contentment, and that is better for you.

Is it permissible for the woman to use her mobile phone while on the street?

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

Modesty should be observed by the woman at all times, so she is free to use her phone while on the street as long as it is done within the limits of Islamic manners. And Allah The Almigty 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.