As far as I know Vigor routers cannot be configured as mesh nodes, only as root. So I'm not clear what this "mesh" connection shown between your routers is? To use the shed router as a repeater for the house router wouldn't you have to use WDS mode, not mesh? If you want to use a mesh connection to the shed why are you using a router there and not a mesh AP?
As an aside, as the mesh backhaul uses the 5GHz band you need a pretty good connection between the house and the shed for this to work in my experience.
Are there any other AP's forming part of this "mesh" group?
Yes, I believe you can use a wired connection between them, but you say that the shed is not wired, so how is that going to work?
Finally, yes you can use the LAN ports on a mesh node, e.g. an AP, to connect non-wifi devices to the LAN, that's what they're for.
Apologies if I've completely mis-understood your questions.
