I first saw Ethel Merman in GYPSY when I was 18, close to the beginning of the run. I saw additional performances both at the Broadway Theatre and later when it moved to the Imperial. Furthermore, I saw the opening of her tour of GYPSY in Rochester, NY when I was in college there. I, too, have heard stories of Merman walking through her performances or "phoning in" performances as the run progressed, but I did not encounter that. She was always the star, giving her all, and acting her part well, not in a "wooden way". I agree with Ludlow's assessment of her performance when he saw GYPSY as a youngster. She was a powerhouse, both as a singer and as Rose, the character.
As for Arthur Laurents describing working with her like training a dog, Laurents did not direct her, Jerome Robbins did. Merman knew that this show was her first chance to really get down to some serious acting, and she followed Jerome Robbins's direction explicitly. She called him "Teacher".
As to answering the specific topic of this thread, others have addressed this well in their posts. Merman had already received a Tony as the leading actress in a musical for CALL ME MADAM, in 1951. And Mary Martin WAS a friend. It was Merman who responded, "How can you buck a nun?" Her real and angry disappointment occured when the movie part of Rose went to Rosalind Russell, who in my opinion was terrible, as was the whole movie, except for Natalie Wood, and Paul Wallace as Tulsa, repeating his stage role.
Incidentally, I saw both Mary Martin in THE SOUND OF MUSIC, and Tom Bosley in FIORELLO, and thoroughly enjoyed both musicals which were both excellent as theatre works. They just were not up to GYPSY's higher plane as musical theatre.
"Madam Rose...and her daughter...Gypsy!"