Learn Hardcore German: Der Haken neben der Tür ist stabil, also lassen wir dort den Schlüsselbund hängen. - The hook next to the door is sturdy, so we let the key ring hang there.

Two-way prepositions like neben take the dative when indicating a static location (where something is) and the accusative when indicating movement toward a place (where something is going). Here we’re describing where the hook is—its position next to the door—so we use the dative. Tür is feminine, and the dative singular article is der, hence neben der Tür.

Why is den Schlüsselbund in the accusative case?

den Schlüsselbund is the direct object of the verb phrase lassen hängen (“to leave hanging”). In German, direct objects are in the accusative. Schlüsselbund is masculine, so the accusative singular article is den.

What function does lassen serve here? It doesn’t mean “to allow,” does it?

In this context, lassen means “to leave something in a certain state or place,” not “to permit.” The construction lassen + object + infinitive can express leaving something somewhere or having something done. So lassen wir den Schlüsselbund hängen means “we’ll leave the key ring hanging.”

Why is hängen used as the infinitive after lassen instead of something like hängen zu?

German’s causative/letting structure is lassen + accusative object + bare infinitive, with no zu. You simply attach the main verb in its infinitive form (hängen) directly after the object.

Why is there a comma before also, and why does also start the second clause?

Here also is a conjunctive adverb meaning “so” or “therefore.” When linking two main clauses, German sets also at the beginning of the second clause and uses a comma before it. Even after also, the finite verb remains in second position:
…, also lassen wir dort den Schlüsselbund hängen.

Why isn’t stabil inflected like an attributive adjective (for example, stabile Haken)?

After a linking verb such as sein, adjectives are used predicatively and do not take endings. That’s why you say ist stabil rather than ist stabile.

What are the genders of the nouns Haken, Tür, and Schlüsselbund, and how can I remember them?

der Haken (masculine)
die Tür (feminine)
der Schlüsselbund (masculine)
German gender often feels arbitrary. The best strategy is to learn each noun together with its definite article (der/die/das). Flashcards or writing sentences can help cement them.

How is the compound noun Schlüsselbund formed, and what does each part mean?

Schlüsselbund is a compound of Schlüssel (key) + Bund (bunch, bundle). In German compounds, the final noun (Bund) determines the overall gender (here masculine), while the first part (Schlüssel) specifies what kind of bundle it is.

Why is the adverb dort placed before the direct object, and could I move it elsewhere?

In German main clauses, adverbs of place like dort commonly follow the subject or appear directly after the verb–subject combination for emphasis. After also lassen wir, you get dort before den Schlüsselbund. You could also say:
also lassen wir den Schlüsselbund dort hängen.
Both are correct; the difference is mainly in what you want to emphasize (the place vs. the object).