Summarized Fatawaa

 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?

All perfect praise be to Allah the Lord of the Worlds. May His peace and blessings be upon Prophet Mohammad and upon all his family and companions.
You can remove some of the letters so that the word isn`t read as (Al-Aziz). And Allah The Almighty Knows Best.

Is buying shared land from someone who has taken possession of it considered halal or haram?

The shared land is possessed by the country i.e. for all people; therefore, it shouldn't be possessed/occupied save by it's guardian/owner permission,thusly it can't be possessed save by having owners' permission. Finally, buying it from a person who don't possess it is impermissible. 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 ….."

What is the ruling on not offering prayer in a nearby mosque, and going to a distant one?

All perfect praise be to Allah the Lord of the Worlds. May Allah`s peace and blessings be upon our Prophet Mohammad and upon all his family and companions.
Praying in the nearby mosque takes precedence unless there was a circle of knowledge, or a larger congregation in the distant mosque provided that the congregational prayer in the nearby mosque isn`t undermined. And Allah Knows Best.