The only bulldogs I've found that look loosely similar are:
Bottle opener:
Door knocker:
I doubt that a hat pin would be a large enough size, but...
Hat pin:
However, none of them are a match. Perhaps the ones on the helmet were inspired by something like this? It seems bulldog heads were a trend once upon a time. To me, the dog heads on the helmet look more like something that could be bought at Mario's Magic Shop on Pee Wee's Big adventure lol.
Regarding the reverse mold, I've read someone's idea on another thread that seemed to be interesting. Username T_bone9600 suggested that perhaps one could make a silicone mold, turn it inside out, fill the backside with something for support, then use fiberglass to copy the inverted mold. Seems like a good idea to me.
Another possibility is that they reproduced it by sculpting a new one while looking at the original in a mirror. I found a reverse version of the flying wall ducks made by a Japanese company but they are NOT identical to the ones on the helmet. There are obvious differences:
Genuine Vintage Set 3 Flying Ducks Japan Retro Chic for Your Wall | eBay
Perhaps the same company that made the ones that were used on the helmet also made a mirror image version of the ducks???
In any case, I would assume that the artists working on the helmet made a mold and recreated the ducks in plastic because I can't see anyone using ceramic due to fragility. Plus, as you've pointed out, it's possible that they have shaped the ducks to the contour of the helmet. In some pictures I see what you're talking about, but in other pictures it seem like it could be an optical illusion (but considering the roundness of the helmet, one would thing these 'flat' ducks would need to be shaped to fit the contour of the helmet, so that's why I'm inclined to believe you are right about that). Another possibility is that they had cut or filed the ducks to fit the contour of the helmet. Who knows. We'll have to figure it out ourselves unless someone is able to contact the artists who made the helmet or the art director. But it's been so many years they might not remember much about it. I've written to Gary Panter (assuming I've found the right Gary Panter) but I haven't heard back from him yet. He's someone who might know something about this stuff since he worked on the set.
NOTE: The backs of ceramic wall ducks are sometimes hollow (the bodies aren't solid). But I've seen others that are solid but the back is flat. I don't know how this info could help, but there it is.
As for the cowboys, now that a clearer image has been posted I think it will help with the identification. But I still can't tell if my theory is right about one cowboy facing forward and the other one facing backwards. I think they must have been custom painted for the helmet. If one is forward facing and the other backwards, they could have painted them to give the appearance that they are both facing forward for the sake of consistency.
Or, if they've managed to make an exact mirror image copy of the mold, then perhaps they've done it with the ducks and the cowboys.
I agree with the headlight thing.