What is the Du`a (supplication) of Istikhara (guidance prayer)?

O Allah, I consult You as You are All-Knowing and I appeal to You to give me power as You are Omnipotent, I ask You for Your great favor, for You have power and I do not, and You know all of the hidden matters. O Allah! If you know that this matter (then he should mention it) is good for me in my religion, my livelihood, and for my life in the Hereafter, or he said: "for my present and future life" then make it (easy) for me. And if you know that this matter is not good for me in my religion, my livelihood and my life in the Hereafter, or he said: "for my present and future life" then keep it away from me and take me away from it and choose what is good for me wherever it is and please me with it."

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 permissible for a woman to adorn herself with kohl (eyeliner) and go out wearing it?

Kohl is a form of adornment, and it is forbidden for a woman to display her adornment to anyone other than her husband or a Mahram. And Allah Knows Best.

 If a disposable water bottle has the logo (Al-Aziz Water Shop) ,is it allowed to throw it away considering that (Al-Aziz) is one of Allah`s beautiful names?