Regarding Rome and the Tonys, the musicals with his greatest scores (Fanny, Destry Rides Again, I Can Get It for You Wholesale) had the misfortune of opening in years in which nominees were not announced or there was unusually strong competition.
In '55, the Tonys named Fanny as one of 6 finalists for Outstanding Musical, but there were no actual nominations. The Pajama Game was listed as the winner.
In '60, The Sound of Music, Fiorello!, Gypsy, Once Upon a Mattress, Take Me Along were the Musical Play nominees. Destry did score nominations for Andy Griffith, Dolores Gray, choreography, and direction. Michael Kidd deservedly won the Tony for his rip-roaring dances.
In '62, Carnival!, How to Succeed in Business Without Really Trying, No Strings, Milk and Honey were the nominees for Outstanding Musical. Wholesale's only nom was for Barbra Streisand. Personally I would have nominated Harold Rome over Richard Adler (he composed Kwamina's score) for Outstanding Composer.