Glad I stumbled upon this.
How can the reader/listener tell this meaning of neH as opposed to "he wants"? For example, could a reasonable reader not also interpret this sentence to me, "He wants to return to his home"?
Context will generally make it clear. But since Duolingo lacks context, on this exercise, we also accept "wants" translations as correct.
As an adverb shouldn't neH be at the start of the sentence?
That's usually where adverbs go, but neH is a very special adverb. When used on verbs it always directly follows the verb it is minimizing. It can also be used directly after a noun to mean "this noun and nothing else".
neH is exceptional in that it follows the verb or noun that it modifies.