Is there a Vim equivalent of Emacs's C-h k followed by the key combo, which basically says, "Tell me what the official command name is for this keystroke?"

I don't know of anything quite like that but the help can be searched by key, though only for the default key combinations, e.g. ":h i^a" would show the docs for the ctrl-a command in insert mode.

Note though that vim really doesn't have commands the same way as emacs does. That's why you generally see assigning one key combination to another instead of some "command name."