1. OSM tiles do not currently render labels for relations
2. No, this was the proposal but I don't believe it is implemented
3. The country place node is far less useful that other nodes, but I would say for consistency should be
included.
Its worth noting that Nominatim was recently (in the last 2 weeks) upgraded to handle duplicate node+relations pairs and to follow and merge data based on the label relation member. Given this my current recommendation would be:
1. Create a node tagged place=* with suitable name/tags at the central location of the admin feature (this may not be the geometric center, but rather the place that would subjectively by thought of as the center)
2. Create an administrative boundary relation with admin_level defining the boundary of the feature in the normal way
3. Add a 'label' member to that relation linking the label node and the relation (nominatim and other data users can use this to deduplicate data)
4. Add a wikipedia tag to either the node or relation to allow Nominatim to work out the importance of the feature
This would be my current gold standard for admin data to best work with both Nominatim and tile rendering while avoiding 'tagging for the renderer'.