I had a really bad burn on a formica counter top so my husband cut it out and placed a ceramic tile trivet in its place. It's really cool looking and serves as a built-in hot spot. To look really good, it needs to be level with the counter top.

I have worked with pottery shards a little bit. The only caution I would make is that not all the pieces will be the same thickness. You'll have to be very careful how you place them. I wonder if rolling with a rolling pin (or similar) would help after you've place them in the mortar? Good luck!

Plates and dishes are made of one type of ceramic, whereas tile is made of another. Ceramic tile is stronger, thicker, and the glaze is designed to be more scratch resistent. Because plates and dishes are generally more fragile, I do not think that plates and dishes would be a good countertop surface. Moreover, it would be next to impossible to get them to be level: much worse than ceramic tile.