Is it really allowed under IKC rules to judge at a show AND also have one of your dogs entered and exhibited at the same show (obviously not in the same class and handled by the co-owner who happens to reside at the same address)?

I highly doubt it!
I would contact the IKC if I were you. Nothing will be done if no one mentions it.
I recall green stars being taken off a family of exhibitors as they tried to get round this rule by changing ownership names, however at Crufts the dogs were all under the same names but a week later at the St Paddys day show they only had 2 of the 3 names they normally had............3rd name was judging at St Paddys then at the next show the dogs were back under all names!

You could be talking about the same dog / people!!!  It is under a set of names to show in the UK, and another set in Ireland.

FCI rules prohibit dogs being shown if anyone who resides at the same address as the dog is registered under ownership as even though the actual judge may bot be an "owner" but am not sure if the IKC has to follow FCI rules being an affiliate country.

> You could be talking about the same dog / people!!!  It is under a set of names to show in the UK, and another set in Ireland.

A dog can not be owned by different people in different countries, i would seriously contact the IKC/UK KC about this as someone is trying to cheat the system.