First perhaps the most obvious potential error. You can plug in the plug either way around, and only one of those ways is correct. The flat side of the plug should be on the same side that has two connections on the sides.
<- Flat side on this side.
Next, there are two modes that the adapter can be in, sync and MIDI send. Sync mode is meant for LSDj and Nanoloop, and only offers synchronization, no note data. MIDI send is only meant for mGB, and does not work with LSDj. You can set the mode with the NLMIDI tool which is a command line tool.
To use LSDj synch, run this tool as nlmidi04 -sync to change the mode temporarily or nlmidi04 -SYNC to save the setting.
Synching LSDj can be done as follow. In the project screen, set MODE to SLAVE or MIDI. I *think* MIDI is the correct mode for use with the adapter. Either mode should work, but one of them might give a very small delay. To start playback, place the cursor where you want to start playing, and press the start button on the Gameboy. You should now see WAIT in the right corner of the screen. Now press start in Pro Tools, and the song should start playing. This of course assumes that Pro Tools has been configured to output synch data to the NL MIDI adapter and not just note data.
To use mGB, run the tool as nlmidi04 -midi to change the mode temporarily or nlmidi04 -MIDI to save the setting. mGB should work right away when you send note data to it. You must send notes on channel 1-5, and not on any higher channels. You are correct that there is no way to audition sounds in mGB without MIDI input.