Whenever using copper as anode, I end up with cyan dust that doesn't dissolve in water.
What's that (presumably) oxide? Is it poisonous? Can I dissolve it in something to perform electroplating with it?
It appears to be copper hydroxide or copper carbonate. Copper oxide tends to be black/grey/brown. Copper hydroxide is blue and copper carbonate is green. Copper carbonate forms when the hydroxide is exposed to CO2.
If you would like to dissolve it, try an acidic solution.