Learn German: Die Frau hängt das Bild über das Bett. - The woman is hanging the picture over the bed.

über is a two-way preposition. Because the woman is actively moving the picture to a new location, there is a clear destination. When answering the question "where to?" (Wohin?), two-way prepositions take the accusative case.

Would it change the meaning if I used the dative case and said über dem Bett?

Yes, it would change the meaning completely! Two-way prepositions with the dative case tell us where an action takes place (Wo?). If you said Die Frau hängt das Bild über dem Bett, it would mean the woman herself is hovering above the bed while she hangs the picture!

Doesn't the verb hängen mean that something is already hanging on the wall?

The verb hängen is flexible and can be used for both the action of putting something up, and the state of it just hanging there. In this sentence, because there is a direct object (das Bild) and a person doing the action, it describes the active movement of hanging it up.

Can I combine über and das into a shorter word?

Yes! In everyday spoken German, über das is very often contracted to übers. It is completely natural to say Die Frau hängt das Bild übers Bett.