The responsibility for the flyscreens can reside with the OC or the lot owner, depending upon who put them in. If the fly screens were an original fixture (put in at the time the building was built) then it would be the Owners Corporation who would look after them. If an owner put them in, then it is that or subsequent owners responsibility to maintain.
In relation to hanging out the window, I would suggest instead going off to the local hardware store to buy an extendable handle window cleaning blush and squeege. I have had same situation in a number of apartment blocks I have lived in, and find using one of these allows me to clean the outside windows in relative safety.