No
No
It is, yes. The Secretary of Education (head of the Department of Education) is a part of the Cabinet.
Education is a department in the Presidential Cabinet.
Yes, it is a Cabinet department
Yes, it is a Cabinet-level department.
That depends on the department. If the department is a Cabinet Level department (Defense, Interior, etc.) the head is a cabinet secretary, while other departments (Postal Service, Fish and Wildlife Department) are not Cabinet Level and therefore their heads are not cabinet secretaries.
The first Cabinet department formed in the 1900s was the Commerce Department.
The Department of Veteran's Affairs was not part of the original cabinet members. It was not until after World War II that there needed to be an entire department devoted to caring for veterans.
The Department Of Homeland Security.
No, since it is a part of the president's cabinet, the president controls it.
The Department of Defense is a Cabinet-level Department in the United States government. As a part of the Government, it is public sector.
As a group, they are part of the Cabinet; separately, they are a Secretary of the Department--such as Secretary of the Treasury, etc.