He can't make a perfect circle without his arm.
The mechanics of Edward's ability to transmute without a traditional circle are not that well defined. He enters the Portal of Truth and then gleans some kind of knowledge about alchemy that allows him to do this. He probably couldn't tell you why he needs both arms, he just knows that he does.
That said, we can try to make an educated guess.
While it's true that he could make a "circle" by putting his arm on his shoulder or other part of his body, this wouldn't serve as a true transmutation circle.
Almost every transmutation circle we see in the series has a single common property: symmetry.
Though this symmetry isn't perfect, with the symbols being different in the circle on the left and Al's Blood Seal on the right having that slightly asymmetrical circle in the middle, for the most part, the circles are perfectly symmetrical, especially the borders and the "structure" within.
If Edward tried to make a circle with one arm, he would have an awkward and asymmetrical circle.
When he uses both arms, though, he is able to make a (roughly) symmetrical circle, and perform a transmutation.
You also mentioned:
... why can't he make a circle between his functioning flesh arm and his non-functioning automail arm?
I don't see why he couldn't.
He never really got a chance to do this, though.
As far as I can remember, the only time his arm was not working, yet completely intact and attached to his body, was right after the fight with the Slicer brothers when Envy and Lust showed up in the 5th Laboratory.
When his arm broke, he was freaking out, and he had also lost a lot of blood, so I doubt he was thinking clearly. Envy knocked him out before he was really able to try anything. Also, his arm breaking was probably a bit for comedic purposes too.