The Red staircase does have red carpet running down the center. It carries with it ceremonial significance, because it leads to the Ceremonial St. George's Reception Hall within the Great Kremlin Palace.
The original was deliberately destroyed by Stalin in 1930's, and was only re-built in 1994.
The term
red doesn't refer to the color of the bricks. In Russian, the square is called
Krasnaya Ploschad. The word
krasnaya means both red and
beautiful while the latter actually refers to St. Basil's Cathedral which is sited at the southern end of the square. This was the original meaning.