Serious question.

So I know limestone is technically a non-renewable resource, since we're using it far faster than it is being created.

Should, uh, we actually be worried? It's like 7% of the Earth's crust; are we actually using limestone fast enough to use up a significant fraction of the usable limestone in the expected lifetime of humanity?

Not in the least. There are other resources that we do need to worry about, of course. But lime and limestone are not on the list.

I've always wondered the same about iron ore, granite, marble etc.

The BBC put together a world resource list a couple years back that I think you might find interesting.

http://www.bbc.com/future/story/20120618-global-resources-st...

It lists a variety of elements (like antimony) that are surprisingly constrained. If I were to pick the most critical material on the list, though, it would be phosphates. The green boom that feeds the world needs phosphates, the world supply of phosphate-bearing rock is small, and there are no possible substitutes.

This. They are not making paper out of stone, they are mixing limestone in plastic. No doubt there are some benefits to it (less plastic is used - though not 80% less because limestone has higher density), but marketing this as "paper" is highly misleading.

Plastics are commonly whitened with Titanium Dioxide and Zinc Oxide already. Calcium Carbonate is a pretty benign chemical as far as potential problems for the plastic waste stream go. I don't see any outstanding issue with its recyclability.