Me, who’s had birds for 10 years and doesn’t have a single memory of this happening with my birds (I feel like I’ve witnessed it once but I’m not sure if my brain is just making it up to mess with me):
Haha, it's not super common I don't think! I've had far more hens mount other hens than cockerels mount other cockerels.
I seem to attract those, I've had a ton of them!
Right now I have an Easter-egger hen who makes this awful, pathetic squawking crow in the dark of the early morning alongside the boys when they start up for the day, and an Ancona hen, OEGB hen, and Cochin bantam hen who all crow occasionally. I feel like I'm forgetting someone else, too, but I'm sure she'll remind me eventually. 
People here are trying to help, just like with any thread. Maybe there was some cattiness in some posts, but we are not trying to be catty or mean in telling you that your bird is not what he was sold as. If someone were to post, for example, a picture of a Dominique and ask people what their experiences with Barred Rocks have been, you'd see people correcting that person about that as well. It's an integral part of this forum. We try to educate each other and correct when we see something posted that doesn't quite add up. I'm sorry if you're upset about it all. I genuinely had no intention of harm in any of my posts. I simply see a young cockerel in your bird, not an older pullet.
That all said, if separating this bird out does prove that the eggs are originating from it, then asking for others' experiences with smaller Cochin hens will likely not help much as in that case this would clearly be a bird with some kind of hormonal imbalance at play, or perhaps even something else like chimerism or some kind of dwarfism. That bird from all appearances should not be laying eggs, so if it turns out that it is, then something else is definitely going on.
For the record, you can sometimes tell who is laying and who isn't without having to wait for an actual egg by inspecting the vent area. A non-laying pullet will generally have a tiny, narrow vent, narrow hip bones, and the vent should be relatively dry, sometimes even flaky. A laying pullet will have a larger, broader vent, wider hip bones, and, well, apologies if the mental image is gross, but her vent will be almost moist-looking relative to a non-laying pullet. On bantam Cochins, the difference in hip width can be hard to tell as they are small and those hips don't widen super far, but those external appearance differences should be there, especially how wide or narrow the vent itself is. Look at one of your older, laying hens' vents to compare and you should be able to see the width difference easily.
I call them the gossip girls
of course they would do that 
Hasn't grown or changed at all with the exception of growing back some picked feathers on its head.
