Looks correct as far as file and BIOS placement.
Did you add in the script?
Here is the condensed installation of the whole thing. It's very straight forward and simple to follow.
The files you sent look like mine as well... only difference I see in the A5200 files is in the Retroarch-core-options.cfg.
I have this:
atari800_opt1 = "enabled"
and I know I did this because I kept getting blank screens and fails. So try that.
As far as the F1/A800 Menu. Try this... get into the RetroPie menu by using "select-X" and see what you have listed in the Quick Menu/Controls/Port 1 Controls. Device type should be Atari Joystick. If not, change it to that.
Also, did you get the correct BIOS files as mentioned. There are several sets that won't work, and also the extra one he mentioned must be installed as well (option 3 on the instructions).
Welcome to the RetroPie world. Some of these emulators are as rkoster put it "tricky" to get going, but once they do... it's done. And you get the benefit of having that old system at your fingertips with a press of 1 button.
The hardest part is usually finding the BIOS files. ROMs are easy to find if you know where/how to look. Tons of them out there in many places. But BIOS files must be specific and correct capitalization.
Try that first and let me know if it works.
I almost forgot... the "lr-hotkeys" Lr stands for Librettro. This is a core that works really well in Retropie for using shaders, overlays and other things. Some of the original Emulators do not allow this and are harder to get working in the RetroPie environment. So the great hobbyists/programmers "forked" many of them into the Libretto core and gave us all the "lr-" based versions.
While running any Librettro core, you have menu options that are separate from the emulator core as well. Accessed as I mentioned above with Select-X or whatever button you have as "X". In those menus you are now working with the actual Libtrettro core and how it interacts with the Emulator Core. So you can add Shaders, Change the graphics size, add Overlays... change controllers etc.. Tons of options.