Those are correct sentences:
The United States National Labor Relations Board is authorized to investigate allegations of unfair labor practices on the part of either employers or employees.
authorize somebody to do something : Napoleon III authorized Haussmann to rebuild Paris.
Under the/certain conditions, a rainbow appears at the end of a rain shower in the quarter of the sky opposite the sun.
"the" and "certain" both are determiners, so we just use one of them. it means: both "under the conditins" or "under certain conditions" are correct.
Addressing themes that were uniquely American, the poet Walt Whitman celebrated the lives of ordinary people.
uniquely/adv. + American/adj ; when "American" is a noun, the correct sentence should be read as " ....Americans"; so here "American" is an adjective.
Though all sedimentary rocks contain iron, the deposits that are richest in iron consist predominantly of minerals such as iron oxides carbonates silicates and sulfides
"Thought....., but" is wrong, wrong and wrong. | 
