Don't worry it takes a little bit to get used to the lingo. Then all the abbreviations and shorthand makes sense.
PSD is the native Photoshop Elements file type. When you click
File> Save as, click on the down arrow next to the Format field. PSD is the first file type on the list.
In the layer palette (and you've got that one located correctly!), you right click on the "background" layer and select duplicate. Then you'll have two layers that look exactly alike in the same document. You can work with one and know that the original is fine. Is it possible that when you're duplicating the layer are you telling to create a new document? When the duplicate layer menu pops up just click okay. Then you should have a Background layer and a Background copy layer.
When you're deleting the invert selection it sounds like you worked on the background layer and NOT the copied layer. When you delete PSE is using your background color to fill in the deleted section of the layer. Once you get two layers in one document this will not be an issue.
Boy would this be easier to explain if we were both in the same room. LOL Keep at it. Let me know if the latest explanations help.