Jim, thanks for the education. I've learned more from you and Noel about what goes on under the hook in Bridge and PS than from all the books I've seen.
"Depending on how you have set your preferences, Camera Raw can be called by first opening Photoshop and then opening Camera Raw."
Then why not do as Noel does and skip Bridge altogether? I wonder how how Noel does that.
Still, what format file does PS use when I start editing in PS?
If you mentioned it, I have overlooked what operating system you are working under. I use Windows. It's possible to see the files in Windows Explorer. And if you want to work that way, all you have to do is double click on a file (if you have your raw/DNG files associated with Photoshop) and Photoshop will start but as soon as it is loaded and realizes you have chosen a raw file it will start Camera Raw. And that is where you make your adjustments to your raw images.
I think the reason you have thought you were editing in Bridge is because if you have configured Camera Raw for full screen there is nothing to indicate that you are in Camera Raw. But there is the little icon just to the left of the histogram that will toggle full screen on/off. And if you go out of the full screen mode you will see at the top in the window bar that you are using Camera Raw.
Personally, when I use Bridge/Photoshop, I like using Bridge because (in my opinion) it offers more viewing options than are available in Windows Explorer. But then that is just my own personal preference.
When a file has been transferred from Camera Raw to Photoshop it isn't in any file format. The pixels have been converted and loaded into Photoshop memory. When you go to save the file Photoshop will give you some choices, but I believe it defaults to PSD which is a Photoshop document.