You can say "cardboard" as the adjective of the materials made from carton

No, I don't think so. Do you mean "cardboard" perhaps?

I think you are talking about recycled cardboard probably coming from cardboard boxes or, if you like, cartons. The phrase "carton-based material" then is grammatically correct, but it is a bit vague, and sounds a lot like "carbon-based material" such as graphene which is used in nanotechnology. I would say "recycled cardboard", "recycled paperboard" or "recycled fibreboard" myself. This is the material starting to be used in eco-friendly housing, for example. While cartons are mostly made out of these materials, they can also be made out of plastic (egg cartons, for example).

For starters, it should have a small letter "c" since it's a common noun. However, will need to have more context on this.

correct