After you have edited a photo in PS, can you record the history and play it onto another image?

New users of Elements (and other image editing softwares) should be warned that there is no such "auto-save" function like in word processors.

To save the editing steps in the full Photoshop, there is a possibility to record 'actions', but Elements can only play, not record such actions.

On the other hand, if you save as jpeg format, there is no way to save layers. And that is the most common way to save steps with different layers if you save in .psd or .tiff format. The layer stack represents the various steps with descriptive layer names. You can show/display layers individually. You can 'stamp visible' the layer stack into a new layer with the shortcut Ctrl + Alt + Shift + E. That is a kind of 'snapshot' in the stack which you can hide/display. That is frequently used to record or illustrate a tutorial.

In my own workflow, I practically always use the ACR module to open jpegs as well as raw files. The advantage is that it is a 'parametric' editing workflow. The original raw or jpeg pixels values are never changed, the result of the edit is not an image copy, it's a 'recipe', a record of the settings of the various sliders. That recipe shows what you have done, and it can be used for other files (last edit setting).

That means that saving the following edits as layers in the .psd format is less frequent.

No.