why is it un séptimo piso and not el septimo piso? Would it be different if the term planta was used instead?

Hola Emanuel

You can say both in Spanish: "Vivo en UN séptimo piso" and "Vivo en EL séptimo piso".

You´d use "un" when you are talking more generally: "on a seventh floor" (in a block of flats) and you´d use "el" when you are being more specific; Imagine you are saying this to someone at  the street and you have the block of flats where you live in front of you. You´d say "Vivo en el séptimo piso" looking and pointing at the top of the building. 

"Vivo en la/una séptima planta" is also correct (but feminine in this case)

Saludos