I do have an "invite" prompt on my SMS contacts that don't yet have Signal: "Invite to Signal: Take your conversation with %s to the next level.". Perhaps it's a function in the beta version that isn't in the live version yet?

I haven't seen a message with a double-tick that's been dropped yet, at least not on a recent version. It can sometimes behave poorly with Android battery-saving mode - however, so does plain SMS. Turning off sync is part of the actual point of battery-saving, but conversely, I never want to not get messages. That's arguably Android's problem, not Signal's. Still, it uses GCM, so at least there's hope for the deep sleep (which I haven't tried yet).