Toasts "pop out," really, though "pop up" makes sense, too. Hence, the general use of the term "pop-up blocker".
I have experienced, and only within web browsers, what could be accurately called a pop-under. A new tab or window (depending on how the browser is set) is opened by another webpage but it does not receive focus at the time. You tend to discover them as you're closing down your browser windows just before shutting down.
Take a screen shot of a couple of the things you're making reference to. It would certainly help with having someone recognize what their source might be.
It would also help if you would do the following, as there are all sorts of gaps that it would be helpful to have filled:
Using Piriform’s Speccy to Collect Your Computer’s Hardware & Software Specifications & Report them on BleepingComputer
1. Go to the Piriform Speccy Download Page and download the program.
2. Run the installer you’ve just downloaded. Note: Speccy free now comes with Avast and the Google Chrome browser bundled with it. Make sure that you UNCHECK the options to install both of these when you see the checkboxes during the Speccy install sequence. When you reach the end, uncheck the View Release Notes checkbox, then activate the Run Speccy button. You will be presented the Speccy Main Window:
3. From the File Menu, activate the “Publish Snapshot” item (or press ALT+F,U):
4. You will now be presented with the Publish Snapshot Dialog:
You will, of course, activate the Yes button.
5. Finally, you will be presented with the Snapshot URL Dialog:
on which you will activate the Copy to Clipboard button so that you will have the snapshot web address to paste into your message.