This answer will probably be a partial one, but I hope it will help you.
Cats are solitary animals. They want their territory for themselves. If another cat is in their territory, the reaction is fight the intruder or move to another place. Your cat doesn't have anywhere to go and is more than scared.
Your cat will normally react to an intruder by scent marking and by not hiding the poo. Not hiding the poo is what can be described as the cat's way of yelling "Get out of here! This is my territory!"
But your cat is going a step further. It is extremely scared and has no more options.
What you, the owner, can do is try to avoid contact with the male cat and you need to remove the smell (of the male cat.) The new cat has scent marked your door and probably a lot of the area around it. To remove this smell you use a weak solution of bleach (chlorine) in a spray can and spray around your door and other areas where the male cat have been(no need to dry it off). If you pet the male cat(hard not to,) wash your hands before you pet your own cat.
Your own cat needs to feel safe when it is using the litterbox, so this has to be moved to a place where it feels safe. For a cat to feel safe, it needs to
be able to see any type of danger (the male cat, even if it is not there in your house.) You might even have to place the litterbox on an elevated place (try to think as a cat) where I can see if anything is coming my way.
The most important thing.: try to make your neighbor get the male fixed(neutred) at any cost This is about the pheromones of the male cat(not only the marking) but also the pheromones in it.
It is not your cat's fault it reacts like it does. It is its nature.
This is not important. when my mother died, I got her cat (Trine) moving in with me, but i had my own cat (Electra.) My mother and I lived in the same house but different floors. Electra did pee in my bed as she was too scared of the new cat to go to the litterbox. This was two cats knowing each other for several years, same house different floors. After two weeks they did sleep close together in the same bed and used the same litterbox, two cats different ages and not close friends but still living together.