No, i am not sure she is kicking him but when i just got them the male didnt go outIf a cat almost got them, then I would try to make it safer where you keep them. Nothing should be able to get at them.
Are you sure she is kicking him out? It's normal for the hen to spend the night in the box and the male normally likes to roost outside the box, but within sight of it.
I am living in dubaiIf they are domestic birds they cannot live in a box tied in a tree. They need some kind of loft or coop where they can be locked in safe at night, and a wired aviary to go outside into the fresh air and sunshine, where you can set up a bath for them to enjoy. They aren't safe in a tree. Ferals live in flocks because they are safer that way. 2 pigeons aren't safe outside. Something will get them, or they will leave.
Where are you located?