Learn Dutch: De wolf let niet op een vlieg. - The wolf pays no attention to a fly.

No, here it acts as a fixed preposition. The verb combination is letten op (to pay attention to). While the separable verb opletten means 'to pay attention', you use letten op when you explicitly state what receives the attention. If it were the separable verb without an object, op would go at the very end of the sentence: De wolf let niet op (The wolf is not paying attention).

Why does niet come right before op een vlieg?

In Dutch, niet usually goes after the finite verb but before prepositional phrases that are structurally tied to the verb. Since op een vlieg is the fixed prepositional object of letten, niet is placed directly in front of it to negate the action directed at that phrase.

Could I use geen to negate the fly and say 'De wolf let op geen vlieg'?

No, that sounds unnatural in this context. Even though geen is normally used to negate indefinite nouns (like een vlieg), when the noun is tucked inside a fixed prepositional phrase attached to a verb, Dutch speakers strongly prefer to negate the entire phrase by placing niet before the preposition.

Is the Dutch verb let related to the English word 'let' (to allow)?