If I play an Exalted Angel as a faced down 2/2 creature, handle turn over, can I then morph it face up in opponents turn?
You can turn it over by paying its morph cost any time you have priority.
If I can, and if I do so, will she be suffering from summoning sickness on my next turn, or can she attack right away?
When you turn it over will have no effect on whether or not it can attack. It's is the same creature, so if it's been under your control continuously since the beginning of your most recent turn, it can attack. Turning it face up just changes (pretty much all) the characteristics of the object, but it's still the same object.
Not sure what Awakening has to do with any of this. :/