You can't have a dative be the subject of a clause. quidquid...movetur is a new clause. It's something like "rule over...[everything] whatever moves on the land."
You don't need to add anything to the Latin, that's just how the language works. Think in English, "I rule over whatever moves." Do we have a prepositional phrase ("over whatever") with an implied subject? Or is it an implied prepositional phrase with an explicit subject? In Latin, it's the latter, because whenever you start a new clause, you need the subject to be nominative (except in infrequent cases of attraction). The subject of movetur is quidquid. The dative is implied.
As far as cuicui goes, you can't trust Wiktionary. Here is the relevant section in A&G (151.b):
In quisquis, whoever, both parts are declined, but the only forms in common use are quisquis, quidquid (quicquid) and quōquō.
[*] Note 1.--Rare forms are quemquem and quibusquibus; an ablative quīquī is sometimes found in early Latin; the ablative feminine quāquā is both late and rare. Cuicui occurs as a genitive in the phrase cuicui modī, of whatever kind. Other cases are cited, but have no authority. In early Latin quisquis is occasionally feminine.
You'll often see cuicui modī written as one word, cuicuimodī. This however is not dative, but genitive.