I would go with a solid roof and utilize it for some storage area. As for ventilation, you mention putting doors on the shed so I assume you will be closing them at night? Maybe during the day, too, at times? If so, I would suggest increasing the ventilation in the shed as a whole. It looks like the eaves are closed but you have cut some holes where the roof peaks. Is that to be your only ventilation? If so, and you plan to keep the whole shed closed at times, you will need more. I suggest installing some top hinged windows that open outwards in all the walls of the shed to let in plenty of light and give you ventilation options.
As for the chicken wire vs. hardware cloth debate, if you are closing the shed door at night then you are probably safe with the current chicken wire, as the shed itself looks very secure. If you might not be there to shut them up every evening or plan on leaving the doors open during warm months then I would definitely consider using hardware cloth over the whole thing.