One should not stop using Venn diagrams, you only (like any other plot) have to use them in the right circumstances! In many cases they are still the most intuitive and easy to interpret plots ...
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lieven.sterck
I don't think there exist any software that will generate this.
Personally I would not go above 5-6 , as those higher numbers become really difficult to interpret (let alone generate). You can consider different types of plots though for higher numbers : Upset plot for instance.
this online one will give diagrams up to 6: Draw Venn Diagrams
I don't think that is possible. You may have some luck if you look for an Euler diagram instead. But as lieven.sterck said, it is just not a good idea to represent such a complex dataset with something venn-like.
7 is the max I've seen, though even 6 sets usually starts to look pretty stupid. I'll second the other recommendations to use upset plots.