Deliberate mistake: The ghosts often walk around furniture even though it was established that they can walk through solid objects.
Suggested correction: Just because they can walk through solid objects doesn't mean they have to. They were used to walking around things in life so just carry on doing so as a reflex action. They often use open doors too, only walking through walls when it's more convenient for them to do so or the doors are closed.
Revealing mistake: The ghosts often cast shadows.
Suggested correction: You can also hear their footsteps. But I'm not sure this is a mistake, just their own perception. Within the ghostly world they appear and act just like they would in life.
The ghosts are manifestations of energy, therefore they have no physical body to block light. Even if in the ghostly world, physics are physics and there should be no shadows.
Character mistake: The Captain's medal ribbon bar is sewn onto his uniform back to front, with the 1939-45 Star on his left and the 1939-45 War Medal on his right.
Suggested correction: This is explained in the final episode, as he took somebody else's medal ribbon bar in order to get into an event only open to "decorated" officers and hurriedly put it on the wrong way round. However, this is also a mistake. Although the Captain never left Britain, he would still be entitled to the War Medal and Defence Medal, so would legitimately have two of the ribbons he wears. And having campaign medals does not make one "decorated" in any case.