i see no reason not to sing with your eyes closed -- people tell me that's how when they know i'm 'on' (when i'm scared, i look off to the extreme left).
the only danger i can think of is swaying so much that you fall, or clinking your teeth on the mic, if you're using one.
On another note, one thing that really relaxes me and opens up my diaphragm, resulting in the best vocal magic, is yoga. One simple exercise if I'm feeing tight: i put my arms behind my back and press my palms together, working them up my back until they rest between my shoulder blades.
It's called Eagle Pose. I don't know if eagles are good singers, but it sure works for me.
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