Hrmm. I'm not seeing the same thing when I do some tests. It might be a subtle difference in versions of unity (I'm testing in 2018.4). Maybe simple fix for now is just adding a pause before the dialog (The scene will still be fading in anyway). Easiest is just a `..` line for a 0.25f sec pause.
As for 1), that's something I've been meaning to add, having text that stays with the character if they're moving. I'm going to want that in my own games too. It's not possible yet though. As for 2) I think an option to disable/re-enable text for a specific character would be useful for background dialog, where you might want speech but no text. Again, that's a feature I'd need to add. If you don't have speech though, you could just check if their position is off-screen and not play the line if they are. eg
if ( Mathf.Abs(C.Dave.Position.x - Camera.Position.x) < 160 )
{
C.Dave.SayBG("Some random background dialog");
}