When Emily is singing in anger at Sera, two of her extra eyes appear; symbolizing she is no longer blind to the hypocrites of Heaven.
Adam doesn't know what he did to get to Heaven. Though considering he's a vulgar, unlikeable jerk, Adam likely got it by default; since both his wives went against the angels' script, but he stuck to what they wanted.
Contracts in Hell work TO THE LETTER. Valentino only has control of Angel when the latter is in his studio; because that's the deal Angel signed. So it's no wonder Valentino was throwing a tantrum when Angel moved into Hazbin Hotel; he had no control there.
Charlie choosing to use Angel Dust as her proof instead of Sir Pentious makes sense. Angel is struggling the most with rehabilitation, backsliding a lot; which is leaving him confused and lost. Pentious took to improving himself off the bat, even genuinely apologizing to Alastor for tearing his coat.
When Vaggie is shown in the flashback to when she lost her eye, it's appropriately shown that she has an empty socket, albeit shadowed out. However, its implied her EYELID was cut out too, which is why Vaggie never takes off her eye-patch; it's the only thing keeping stuff out of the socket.