Copper is available in all kinds of thicknesses. My local restaurant has tables that are made of pretty thick stuff and not dented. It's probably 1/16" or more. You would need tools and force to dent it.
Copper Sheet, Copper Flashing, Copper Sheets, Copper Foil Rolls, Copper Sheeting for Arts and Crafts and Various Applications
These guys seem to have it all including a thickness guide with a video showing you the flexibility of different thicknesses.
As for the toxicity issue... remember everything is toxic. The poison is in the dose. The "toxicity" is actually a plus because it is antibacterial. So the salmonella from the raw chicken I put on the counter might not linger there if I don't clean it up well enough.
As for the taste issue, France (where they KNOW food) has lots of copper pots in use for various specialty cooking applications. Most of the time though I would be cutting food on a wooden cutting board anyway so if I set my food on the counter for a minute or two I don't think it'll affect the taste.