For the second one, if you want to keep the "... is out of..." construction, my suggestion is:
"My phone is out of power."
But in regular (informal) conversation, I'd more like just say "my phone is dead."
As for "did it up," and "out of battery," these are awkward from the perspective of American English. So it looks like we can all agree on one thing, at least: these are good examples of differences between American and UK English(es).