No, it isn't a resonator. It's a gizmo that fits on the back of the mando holding it away from your body. I'll try to remember what it is and report back (unless, hopefully, someone beats me to it!) Apropos of headphones, surely the point of a session is to interact with the other musicians. I can't see headphones exactly assisting with that.

As others have said, be aware that the presence of any amp for any reason in a session is almost certain to be unwelcome, very likely expressed vociferously. You may be able to carefully nurture its acceptance in your own local session with your mates over time. That's what I did for my poor little harmonicas, but I still wouldn't dream of showing up at someone else's session with my amp. Recently, I've discovered harmonicas (and techniques) that are much better at cutting through and I always prefer to ditch the amp if I can. I don't agree that mandolins are that good at cutting through, though they do sort of add to the mix (isn't that what you want?), but you'll never convince me that a guitar ever needs boosting, unless your session is the size of the London Symphony Orchestra.