Is Django bullet-proof or immortal?

Short answer: No

The most obvious explanation of that scene is that the shells were casings from his own bullets or spare ammunition he hadn't been able to load before Stephen (the black estate manager) captured Broomhilda and threatened to shoot her. His action preceding that scene involved sheltering from hostile fire, which he wouldn't need to do if he were bullet or injury proof. And he cast off his coat because he wanted to show he was unarmed.

Earlier scenes also suggest he is not in any way proof against injury. In the early scenes he shows the scars from being in leg irons. In the scenes after he is captured he fears being branded or castrated. His fear certainly suggests he doesn't think he is immune to pain or injury in any way.

And there doesn't appear to be any other hint that he is special elsewhere except that he is both intelligent and a very fast and accurate shot.

I think what falls to the ground are the big oblong buttons on his coat that are normally found on Western type of jackets. Another theory is that somehow his jacket was bullet proof and those were bullets he was hit with falling out. Unlikely but just a theory.

They appear to have tapered ends maybe twisted. That could be a black powder paper load with ball and powder or maybe a button. There were no cartridges used in those colt cap and ball revolvers.