Petseta sounds quite similar to "Peçete" in Turkish. But we use it in a different meaning :)
Both ντους and Dusche are from the French word "douche", which means shower. Greek has many loan words from French and Italian :)
Is it also the word for the room where the shower is ? (Like it is in French for instance)
It can be. The word for the room is usually μπάνιο, if it's in a house and also has a toilet and a sink, but if it's just a shower room in, say, a gym them "shower room" is ντους. Or ντουζ, I have to add.
Is there any logical reason why "The towel's in the shower" should not be accepted? To me it's the same thing, but DL seems not to agree with me & marked it as incorrect. :-(}