First, I'm very sorry for your loss of Luna, especially in those circumstances. It's always sad loosing a beloved hen but a young pullet just starting to lay, like this, is especially hard. I've experienced like many chicken keepers the feeling of helplessness and guilt that you may be going through. It's important to be responsible, and to try doing better, but we also to have to acknowledge that sometimes there is nothing we could have done even knowing more.
I'm by no means an expert on prolapses but I kept ex-batt's for four years and a half so I've had some experience with reproductive issues. Here is my answers to your questions. And no questions are stupid, by the way. You'll see that we won't all answer exactly in the same way. It's always necessary to question things you read on the web about chickens, unless they actually come from a veterinary website (and even then).
The most likely explanation is that the prolapse killed her.m, not the bath. But I would also refrain from soaking or bathing a hen in great distress unless she was used to it and you know she enjoys baths. I know that is an advice often given for egg bound chickens because the heat is supposed to make their muscles relax and push out the egg. But many chickens hate baths and will actually stress from them. If it's really cold outside, I would eventually put a warm bag near them, but even that can be a stress.
I personally don't think calcium would make a difference. I know the usual advice is to give human calcium citrate pills in case of trouble laying or actual egg binding. I've done that religiously for years since my first ex-batt was egg bound. I still do it now for one of my hen. It has never, ever, made a difference or actually helped a hen pass an egg, even when I gave huge amounts. I don't believe calcium deficiency would have caused such a sudden issue. Hens are able to take calcium from their bodies to make eggs especially when they are young ; the deficiency will show itself gradually through weaker shells.
That said, my ex-batt's never took to eating crushed egg shell or crushed oyster shell. All my other hens do. As long as I had my ex-batt's for this reason I made sure layer feed was always available in one of the feeders on top of the fermented grains and chick crumble mash I feed my chickens.
I would answer no for the reason I explained above.
I have only seen two prolapse and got them back in because they were just beginning. From what I have read, but have not experienced myself, here is my thought. An advanced prolapse is a veterinary emergency. It doesn't hurt to try to get it back in because otherwise the chicken will die anyway ; but the chances of success are slim.
Temperature extreme could be lethal ; I'm not sure it could happen in the way of a prolapse. Extreme heat does impact laying, that I know. I would get a thermometer and check the temperature in the run compared Tou outside. You possibly may need to prepare more cooling measure if you get another heat wave !
The eyes can remain open in death. Often while the chicken dies it will thrash around like it has a seizure - I believe this is due to the nerves. There can still be movement after the chicken has actually died, and I've read it can last for a few minutes, although the chickens I've seen dying stopped seizing after 30 seconds to a minute. Liquid may come out of it's mouth and vent, and sometimes feathers may fall. I check for breathing ; I am not sure about using a stethoscope, as I've never done it.
One last thing I want to say. One of the possible cause of prolapse is an egg too big, and that is an issue one reads about with Australorps. Again, this is something I read but can not vouch for ; but I think it's a possibility, as hens that are just beginning to lay can lay eggs that are very small, which is the most common, but also huge. It was the case with my hen Vanille who laid huge eggs her first month, had reproductive issues all her life, and died from being eggbound before turning three.
I hope you will get more helpful answers, and gradually cope with the loss of your Luna.