It appears that the standard gamepad for FS-UAE is the Dualshock 3. Good that you decided on a commonly available product.
I have tried the latest FS-UAE with Apidya, mapped my controller using Gamepad/Joystick setup in FS-UAE Launcher, and it still makes no sense to me at all:
Why are the buttons not used for the various functions listed at http://fs-uae.net/keyboard-shortcuts?
Apparently the only buttons used by the emulator are
Menu = Toggle emulator menu
Cross (or ‘South’) = Accept menu item
Triangle (‘North’) = Amiga fire button
You mentioned that Left Trigger (L2) toggles autofire. Well for me it doesn’t! 
Pushing various buttons cause a joystick event to be sent. For instance, Circle makes Apidya bee fly DOWN.
I checked C:\Users\marco75\Documents\FS-UAE\Controllers\ps_to_usb_convert_cable_12_4_1_0_windows.conf and everything is mapped as it should be to account for the characteristics of the adapter, so I’m at a loss.