What dangerous gases could electrolysis with water and sodium carbonate produce?
My friend pointed out that it also contains chlorine and that's what he's afraid of. 


The amount is laughable at. However, using distilled (deionized) water is recommended for other reasons.


  I failed to find out whether these compounds are hazardous. They're in solid state anyway.


They are reasonably harmless. Of course, eating them is not recommended, but copper compounds have low toxity. Move to graphite or platinum electrodes is still recommended.