Ein Bild vom Vorjahr hängt an der Wand.

The preposition an is a Wechselpräposition (two-way preposition).
• For location (answering where?), it takes the dative case.
• For direction or movement (answering where to?), it takes the accusative case.
Here, an der Wand describes a static location (“where is the picture?”), so die Wand becomes der Wand (feminine dative).

Yes, both are grammatically correct.
• Ein Bild aus dem Vorjahr emphasizes origin by using aus (“out of the previous year”).
• Ein Bild vom Vorjahr (“from the previous year”) is more idiomatic and very common in everyday German. The nuance is minimal, and speakers use them interchangeably in most contexts.

Absolutely. des Vorjahres is the genitive form (“of the previous year”).
• Using the genitive feels more formal or literary and appears often in written texts.
• Using vom Vorjahr (dative with von) is more colloquial and frequent in spoken language.

An der Wand implies attachment or hanging on a vertical surface, which fits pictures.
Auf der Wand would suggest something lying flat on top of the wall’s horizontal surface—unusual for a hanging picture. For objects hung or fixed against a wall, an der Wand hängen/sein is the standard expression.

German makes a clear distinction:
• hängen = to hang (objects suspended or attached vertically, like pictures).
• stellen = to place/stand (objects set upright on a surface, like a vase on a table).
• legen = to lay (objects placed flat, like a book on a shelf).
Since the picture is hanging on the wall, hängen is the correct choice.

Yes, you can front the location:
An der Wand hängt ein Bild vom Vorjahr.
This shifts emphasis onto where the picture is, but the basic meaning remains the same. German word order allows such flexibility to highlight different parts of a sentence.