Not sure what this is. A rimfire with no rim?
cartridge length 8.42mm
case length 5.46mm
bullet 5.70mm
case diameter 5.70mm

This one looks like a percussion cap that was fitted with a round ball (shot). I have seen similar “cartridges” made-up with different percussion caps, some having rare markings.

A pinfire with the pin missing.
Is there a hole in the side of the case near the base?

Definitely percussion cap.

Yes, this is “big” percussion cap. I have similar small cap with flying bird too.

Thanks guys.