Learn Italian: Tutte le stanze sono aperte, tranne l'area riservata. - All the rooms are open, except the reserved area.

No, tranne directly precedes the noun and its article. That is why it goes straight into l'area, without needing an extra preposition like a, da, or per in between.

Could I use eccetto instead of tranne in this sentence?

Yes! Tranne and eccetto mean the exact same thing ("except") and are perfectly interchangeable here. You could easily say ...eccetto l'area riservata.

Why is it Tutte le stanze instead of just Tutte stanze?

In Italian, whenever you use a form of tutto (all) before a noun to mean "all of the", you must include the definite article. So it has to be tutte + le + stanze (all + the + rooms).

Why does aperte end in -e?

Aperte is the past participle of aprire (to open), but here it acts as an adjective describing le stanze. Since stanze (rooms) is feminine plural, the adjective must change its ending to match, becoming aperte.