Don't have superuser accounts at all if you can help it.
Run Postgres in a limited user. Something that can't access any file or execute any command (like wget) it doesn't need, can't do chmod +x. Can't run a shell. Don't know if postgres needs that.
No, postgres can't work without accessing files or having a working shell (latter kinda works in some limited configurations).
And I don't think it's a reasonable idea to not have a superuser at all. But you can have it password less and only accessible from the local machine and a specific account (eg root).
If the image is not lossy compressed (even if it is for the really clever) you can use a variety steganographic tricks to hide any data you want inside an image. Of course you have to have code to decode it.
Would the payload still be there if the image was rebuilt with Imagemagick?