Ego noctu cum psittaco ebrio saltare soleo.
Usually acts as a verb in these sentences?
In the Latin, yes. The verb solēre means 'to be accustomed to', 'to tend to', 'to be in the habit of', etc. That is where Duolingo is getting the 'usually'.
"I usually dance by night" isn't allowed for some reason...
...said the doll at the temple...
Why not : "i use to dance at night" ?
That sentence makes it sound like you don't dance at night anymore. The Latin implies more that 'I am used to dancing at night'.