To address your list:
- a high quality 3D structure: this you can easily get from PDB, using the answers to the question you linked as starting point. However, it is become increasingly clear that intrinsically unstructured proteins also play important roles in the cell, and for these you won't get a good 3D structure.
- known activity in vivo / known associates in its activities in the cell: interaction partners can be found in databases like STRING, although just knowing the partners doesn't mean we understand what's going on :)
- kinetics of its activity: I know of no good resource on this one except BRENDA for enzyme kinetics. But of course there are also kinetics to protein–protein interactions that would be interesting to know. (Update: There is AffinDB.)
- regulatory elements for expression: there are databases like TransFac, but ChIP-Seq experiments show that we only know a tiny part of the regulation.
But, I think the key to a good answer lies here:
(probably most importantly) a general consensus among experts that it is well described
I would check how often the genes are described in the literature. The GeneRIFs from NCBI may be a good starting point, because these are snippets from the literature that are curated and shown on the NCBI Gene pages. E.g. p53 has >3800 of these. The length of the summary could also be a good proxy. (Or, the length of the Wikipedia page?)