Learn Hardcore German: Der Besitzer parkt das Auto vor dem Haus. - The owner parks the car in front of the house.

Der Besitzer is the subject of the sentence and therefore in the nominative case. German subjects appear in nominative; Besitzer is a masculine noun, so it takes the masculine nominative article der.

Why is das Auto the direct object, and what case is it in?

das Auto is the thing being parked, so it’s the direct object and marked in the accusative case. Auto is a neuter noun, and neuter remains das in both nominative and accusative.

What case and function does vor dem Haus have, and why is it dem rather than das?

vor dem Haus is a prepositional phrase expressing location (“in front of the house”). The preposition vor takes the dative case when indicating a static location. Since Haus is neuter, the dative article is dem (nominative das → dative dem).

Why is the verb parkt in second position?

German main clauses follow the verb-second (V2) word order: the finite verb (here parkt) must occupy the second slot. Whatever comes first (subject, adverbial, etc.) pushes the verb into position two.

How would you ask “Where does the owner park the car?” in German?

Use the interrogative wo (where) in first position, keep the verb second, then the rest:
Wo parkt der Besitzer das Auto?

How would the sentence change if the action described movement toward the house instead of parking (stationary location)?

For movement toward a location, vor takes the accusative case. You might also choose a verb like fahren (to drive). Example:
Der Besitzer fährt das Auto vor das Haus.
Here vor das Haus answers “Wohin?” and uses accusative das.

Why is Haus a neuter noun and not feminine or masculine?

German noun genders often have to be memorized. Haus is neuter by convention (Old High German hus). There’s no suffix rule here, so you learn das Haus as a fixed pairing.