Kirsten, a Merry Christmas to you too, also to Luna of course!

It great that Luna has ajusted well to the new apartment, though it is a pity that you have all these new worries about her safety. I am not sure what, if anything, you can do about the fear of her falling through the bannisters, I know I am a little fearful for Ebby too when she is peering though mine and I don`t think they are as far up as yours are. I suspect other experienced cat owning Pet Talkers may have some answers for you though.
As for the other thing with your landlords I know I would be as worried as you are. Emergency access I could understand but, nice or not, the anytime access would trouble me even if I didn`t have a cat, it doesn`t seem right somehow, especially as you are only a few minutes away. If you cannot/may not get the locks changed could you maybe get an external security keybolt fitted and a notice asking them to call for access. At the very least I would have a notice to be careful of the cat put up. I don`t know what else to suggest. I am sorry that this worry has marred the pleasure of your new home, I hope you can sort something out soon to put your mind at ease.

Best wishes
Chris