I shall try and give you some answers until somebody with more knowledge can give more accurate answers.
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Yes, unnecessary effort as you could leave the the file in ~/Downloads if you like. However, I would suggest moving your AppImages to ~/bin to keep them in one place and help you with your issue 4. appimaged also checks the ~/bin directory which will help.
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AppImages can be stored wherever you like, but a directory that appimaged monitors is preferable if you want to use appimaged to keep things simple.
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Version 1 AppImages did have the problem that they had a popup dialogue that offered to install a desktop icon file but there way no way to uninstall. However, this approach has been deprecated and version 2 AppImages do not offer to install themselves. If you are using appimaged, then deleting the AppImage file will uninstall it from your system.
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I do not know of any way to list all the ‘installed’ AppImages. The best way that I can think of doing this is to use appimaged and move all of your AppImages to ~/bin.
Basically, have a look at appimaged and see if you think it helps:
Cheers,
Chris