It’s possible to do what you’re talking about by totally overriding the view, at least the UI part of it. That’s going to be annoying to keep up with, as any changes we make to the “normal” controller/view/js/etc. could affect your module. So I probably wouldn’t recommend it.
That interface is pretty much run on JS, so it might be better to go that route to alter the UI rather than changing the actual markup. This is one of the more complex and fundamental pieces of the UI and data model for S, so it’s relatively complex to add things here outside the handful of “blessed” areas for extension we provide (like data types).
Another user has done something somewhat similar that you might look to for inspiration: New Relators Module (Looking for input) : they are adding extra data alongside regular values.
If your extra info is really associated with the properties and not the values then you could go a totally different route and not add your functionality to this view at all, but that would only work if the “extra data” is the same for a property no matter which item it is used in… I think from your description that’s not exactly what you want.