1. There are curtains on the window. This sounds OK to me. It is the only one which suggests a natural position. 'On' suggests that they are connected i.e. hanging from a rail attached to the window. 2. There are curtains at the window. This does not sound very natural to me. 'At' is quite vague, and doesn't seem to suggest that they are attached. A person could be 'at the window', for example if they in a position where they can look out of the window and observe what is happening outside. 3. There are curtains by the window. This is correct, but has a different meaning. It suggests that the curtains are somewhere near to the window. Perhaps I'm asking someone to hang up some curtains for me, and I'm telling them that the curtains are in a pile close to the window. In the real world, we'd probably say 'The window has curtains' e.g. "The bedroom window has curtains, but the kitchen window has blinds'.

I agree with Su.Ki. that "on" is the only version that sounds natural. I can imagine the phrase "curtains at the windows" occurring in a piece of literature, a poem, or a song, but I think this expression would sound odd in everyday speech.

all correct. They describe different locations. 1= on the window or hanging on/at the window. 2 = hanging on/at the window, or it could mean by the window. 3= not hanging but by near close to the window. all three are a little bit interchangeable, depending on the person speaking and their preference in the way they phrase it.

'at the window' is best. 'on the window' is okay, but it sounds like they are touching the windows instead of just near them. 'by the window' is incorrect.

the first 2 sentences are correct