How does David maintain a secret cabin in the ship without any human knowing about it?
As per Paul D. Waite's comment, and confirmed by the script directions, this is the ship nursery, nothing secret.
The gentle snow falls.
David strides in.
The Colonists. Tier after tier after tier of them in hypersleep.
Also the rows of embryos.
What is secret is what he is hiding among the other embryos:
He coughs up two Xenomorph embryos and puts them in a sample tray with the human embryos.
In the original shooting script, it was instead within the Wardroom.
The Wagner continues as he enters.
He goes to the vertical garden that soars up along one wall.
To a growth of ferns. Kneels and looks beneath them. Smiles.
A neat little row of three very small ALIEN EGGS.