Why does Sergeant-Major Morris throw the paw onto the fire, and who gave the paw to Mr.White?

Nobody gave the paw to Mr. White.  He grabbed it off the fire himself when Morris threw it there.

Morris threw it there because he did not want anyone else to have it.  He knew that any wishes that someone made would be granted, but they would probably be granted in a very bad way.

Because he knew that, he wanted to just burn the paw and be done with it (I don't know why he hadn't just thrown it away before).  But instead, Mr. White grabbed it.

Just as Morris predicted, the White family soon wished that Mr. White had not gotten the paw and had not made any wishes.