Yes, "βραδινό" can also be dinner. Depending on how full and / or how formal the words change. "Δειπνο" sounds more formal.
It is thee-pno, δείγνο would sound completely different. If you have trouble with the audio here, you can check the pronunciation in other websites as well, for example here δείπνο-1 here δείπνο-2 (the first is very clear, the second not so much in this case but better recordings may be available for other words). You can also try both δείπνο and δείγνο here to see how different they sound.
I hope Duo can fix it's own audio when necessary, instead of depend on links at other sights. I also hear a g (γ) sound. Maybe there is a missmatch between the text and the audio on this one?