If the King makes it to the eighth rank and then back to the first rank, can he promote a piece like a pawn can when it reaches the eighth rank?
No. The whole point of the game, the way you win, is by checkmating the opponent's king. If your opponent could exchange the king for another piece then you would no longer be able to win.
In any case, all the pieces except the pawns can move backwards. Without promotion any pawn which reached the 8th rank and couldn't promote would no longer be able to move. That doesn't apply to all the other pieces.
FIDE, the international chess federation publishes the laws of chess on their website. The first 5 articles describe the rules for general play. The articles from 6 onwards describe additional rules which apply to official competitions.