re: Does LSU have at least one of each?

No but somewhat close.

National Awards - How many LSU has [who has won]
Heisman Trophy - 1 [Cannon 1959]
AFCA Coach of the Year - 3 [Dietzel 1958, McClendon 1970, Miles 2011]
AP Coach of the Year - 2 [Saban 2003, Miles 2011]
Bednarik Award (Def. Player of the Year) - 2 [Peterson 2010, Mathieu 2011]
Biletnikoff Award (Best WR) - 1 [Reed 2001]
Draddy Trophy ("Academic Heisman") - 1 [Niswanger 2005]
Eddie Robinson Coach of the Year Award - 2 [Dietzel 1958, Saban 2003]
Home Depot Coach of the Year - 1 [Miles 2011]
Liberty Mutual Coach of the Year Award - 1 [Miles 2011]
Lombardi Award (Lineman of the Year) - 1 [Dorsey 2007]
Lott Trophy (Def. Player of the Year) - 1 [Dorsey 2007]
Manning Award (Best QB) - 1 [Russell, 2006]
Nagurski Award (Outstanding Defensive Player) - 1 [Dorsey 2007]
Outland Trophy (Best Interior Lineman) - 1 [Dorsey 2007]
Paul "Bear" Bryant Award (Best Coach) - 1 [Saban 2003]
Rimington Award (Best Center) - 1 [Wilkerson 2004]
Thorpe Award (Best DB) - 2 [Peterson 2010, Mathieu 2011]
Walter Camp Coach of the Year Award - 2 [Stovall 1982, Miles 2011]

HAVE NOTS
Butkus Award (Best LB) - 0
Maxwell Award (Nation's Best Player) - 0
Mackey Award (Best TE) - 0
Lou Groza Award (Best PK) - 0
Ray Guy Award (Best P) - 0
Davey O'Brien Award (Best QB) - 0
Doak Walker Award (Best RB) - 0
The Sporting News National Coach of the Year - 0
Walter Camp Award (Nation's Best Player) - 0
Woody Hayes Trophy (Best Coach) - 0