It does not matter whether you put a hut in the cage or not, eventually your birds are going to evidence sexual behavior; hens will lay eggs, whether fertile or not, and cocks will proposition the hens for sex. If you want them to have a hut to play or sleep in, give it to them.
The purpose of all life is to reproduce, to create replacements for themselves to carry on the species after they are gone. It is why hens and cocks are different, why birds grow up and become adults and start acting sexual. It is the fact that life comes from life and is self perpetuating, so it is IMPOSSIBLE to avoid. All we can do is allow the birds their lives, allow them to go through the cycles of their lives and control their population as best as we can.
My birds pair off, mate, produce eggs and then I abort the fertile eggs by refrigerating them. In twenty years I have limited my cockatiels, lovebird and budgie populations in this manner and it has worked rather well. Quit obsessing about if something will set your birds off laying eggs or mating; the hormones WILL develop and the birds will act out their cycles of mating and brooding, despite everything you try. Roll with the punches and allow your birds to be happy.