Can we have statistics for a single country only?
The main taginfo site (https://taginfo.openstreetmap.org/) does not do this because it would add huge amounts of data that need to be created, stored, etc. But you can setup your own version of taginfo for your favourite area and add it to this list.
There are more than 100 million distinct tags, but "only" about 75,000 keys. It is simply not feasibly to generate all the different types of statistics, maps, etc. for all tags.
Can we see all tag combinations and not only some of them?
There are more than 100 million distinct tags, doing statistics for all combinations is not feasible with the current architecture and reasonable hardware.
I think tag X should appear in taginfo
Taginfo automatically shows tags that appear in the database, in the wiki or some other sources. You do not need to contact anybody to have your favourite tags appear in taginfo. See also Proposed features.
Yes, it would be. But taginfo doesn't store this information to keep its database manageable. You can use other tools, mainly Overpass API, to find those objects. Sometimes keys/tags that you see in taginfo are not found via Overpass. Most often this is because somebody else just removed this object. Taginfo is only updated once a day, Overpass is continuously updated every minute, so this can happen easily.
Thoses keys/tags have at least one wiki page, so taginfo still shows them.
No points are shown on the map for a key/tag, but the key/tag exists. How is that?
Taginfo doesn't show anything on the map for relations, because (in general) relations don't have a "location". So if a key/tag is used only in relations, the map is empty.