Yes, it is quite well known term, and it means changing the photo or document in a way to make it misleading, but not necessarily with photoshop. See eg: https://www.indexoncensorship.org/2017/08/commissar-vanishes/ If you are editing a photo for a reason other than trying making it misleading, you would never say that that you are "doctoring" the photo.

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Yes: "to doctor something" is a slang way to say "to alter something". Depending on context it can suggest improving something, but more frequently it suggests trying to hide inconvenient facts or trying to mislead others.

I've never heard this before as a native speaker, so I would say it is not common at all:)

You're right. It means that they will alter or change the picture with an editing software.