I love the idea of making boxes like this. There seems to be several you tube videos on how to make them, although probably practise will make perfect!
This one is part 1 of a series that goes into some detail on how to make a round hat box
I think the weight of paper will depend on what you want it to contain. That seems pretty obvious - if you need it to contain heavier objects then thicker card is needed. I guess there is a limit to how thick the card can be so you can still make the curves.