Chris Sims has done a great job over the filling in the origins for many of the items shown, but I noticed this:
I can't place the Penguin, the giant Joker and Two-Face Heads, or the 8-ball.
I can only cover the giant Two-Face head for now, although I suspect I will find the other items eventually. The Penguin (in this case a stuffed bird, not the villain), for example, was one of the first objects ever shown in Batman's trophy collection in Batman #12, so we know it's got to come from an earlier story than that one.
The giant Two-Face head comes from Batman #50's The Return of Two-Face! The climax of that story has Batman and Two-Face battling it out atop the giant bust:
Batman explains the origin of the bust to Robin:
This appears to refer to the initial appearance of Two-Face, in Detective #66:
Although you will note that the bust depicted in that story is much smaller than the one in Batman #50.
The bust made a significant appearance in the Silver Age. Batman #108 (June 1957) included Prisoners of the Batcave, where Batman and Robin are trapped in their hideaway, just as a package they received turns out to be a firebomb. Here's the action as seen in that story:
The air inside the hollow bust is quickly used up and the fire dies out, although not without damaging the bust significantly. The material of the bust is described in the text as glass, although there is little doubt that the Golden Age bust was intended to be of stone or plaster.