Despair is exactly lack of hope (at least it is one of the meanings). 'Spero' means 'I hope' in latin and 'de-' as you probably know means mainly 'without' &c.
Frustration is a feeling. Stress causes frustration. A frustrated person is irritated and impatient.
To be in distress means to be in trouble (a damsel in distress can be rescued by a knight).
Depression is a mood. Depression (mood) - Wikipedia, the free encyclopedia
I think distress can also mean the same as depression.
But I'm not a native speaker.