https://linux.die.net/man/1/gnome-screenshot https://docs.fedoraproject.org/en-US/quick-docs/proc_setting...
I believe it can do all of the things you want. Certainly area capture, remembered area capture, fullscreen capture, all bound to different hotkeys. Mine saves with the name = the timestamp but you can probably config it to be an incrementing index. It's incredibly full-featured.
I also have hotkeys for "capture current pixel's hex code" and "measure bounded box in pixels." When you take a capture you can also annotate it including showing labeled steps. After capture you can do one or more of: save locally (to one or more places), upload (to one or more hosts), copy to clipboard, etc. That includes pastebin if you have text saved to your clipboard so I use this for that also.
Basically I bound a key combo to a script, and that script would call scrot (the app), screenshot, and save the picture to date_plus_random_4_digit_number.png.
I think I wrote that script in Ruby and put it in .local/bin. But you could write it in another language too.
Make two to distinguish between "save entire screen" and "let me drag a selection that it will save".
I want to press printscr or some shortcut and immediately screenshot an area of the screen or the whole screen and by default have it saved to a numerically incrementing files under a folder I configure and not have to close or interact with any popups. This is on Linux. Any recommendations?