Don't worry, either one is close enough (but it's writeable and closable). The spelling here has to do with pronunciation and vowel sounds/rules.
"next" is usually an adjective (something that applies a characteristic to a noun) and since "move next" here probably means move the next something, it's adjectival.
Adverbial use of next would be like in chess: "You move next." Sometimes next is part of a prepostion: "next to"
The verb form of "iterator" is iterate.