Hi @ogogon,
I found myself in the same situation and did some digging, so here is a recap of all the steps.
I'm on Ubuntu 22.10.
From Where should I put .Appimages files? - Ask Ubuntu, I chose to move it to ~/Applications (create dir if it doesn't exist yet).
mv ~/Downloads/arduino-ide_2.1.0_Linux_64bit.AppImage ~/Applications/
These can be solved by executing the AppImage and looking into its mounted folder.
I found explanations here : https://linuxconfig.org/how-to-create-an-integrated-application-launcher-for-an-appimage-file-in-ubuntu
Open the AppImage (don't forget to make the file executable and install libfuse2 as mentioned in the Arduino documentation).
While it's opened, type in a terminal : mount | grep AppImage
This should output something like this :
arduino-ide_2.1.0_Linux_64bit.AppImage on /tmp/.mount_arduinkRI6cP type fuse.arduino-ide_2.1.0_Linux_64bit.AppImage (ro,nosuid,nodev,relatime,user_id=1000,group_id=1000)
Note the path to the temp folder, here /tmp/.mount_arduinkRI6cP (you will need to replace this path in the following commands).
This folder should already include an icon and a .desktop file.
For the icon : find /tmp/.mount_arduinwozLsj -iname "*.png"
This should return a lot of files, for me the icon was conveniently at the end so I didn't have to scroll : /tmp/.mount_arduinkRI6cP/usr/share/icons/hicolor/512x512/apps/arduino-ide.png
Now, copy this icon somewhere, it can be the same folder where you put the AppImage, but I personnaly put it in .local/share/icons : cp /tmp/.mount_arduinkRI6cP/usr/share/icons/hicolor/512x512/apps/arduino-ide.png ~/.local/share/icons/
You can set the icon of the AppImage using its Properties.
Also find the .desktop file (this will add the app to the launcher) : find /tmp/.mount_arduinkRI6cP -iname "*.desktop".
Copy it to /usr/share/applications, or ~/.local/share/applications : cp /tmp/.mount_arduinkRI6cP/arduino-ide.desktop /usr/share/applications/
Now edit this file with your favorite text editor (gedit, nano, vim if you know how to get out of it 
) : nano /usr/share/applications/arduino-ide.desktop
You need to edit Exec and Icon paths to match those where you copied the .AppImage and the icon. You can also edit the name, comment...
Here is what my .desktop file looks like (I also copied bits from the Arduino 1 IDE already installed on my computer) :
[Desktop Entry]
Name=Arduino IDE v2
Comment=Open-source electronics prototyping platform
GenericName=Arduino IDE v2
Exec=~/Applications/arduino-ide_2.1.0_Linux_64bit.AppImage
Icon=~/.local/share/icons/arduino-ide.png
Type=Application
Terminal=false
Categories=Development;Engineering;Electronics;IDE;
Keywords=embedded electronics;electronics;avr;microcontroller;
X-AppImage-Version=2.1.0
StartupWMClass=arduino ide
Save you edits and you should now see the app in your launcher !
EDIT : added line "StartupWMClass..." to fix icon missing at runtime : unity - Icon missing when application is launched - Ask Ubuntu