There are plenty of furnished apts. being offered up in Tung Chung, on a month to month basis, in the $8-14k range. You can walk from Tung Chung to Cathay in 25 min or take a bus in 5, and the family can stay with you for visits until you find something better. I had a 2 bedroom and it worked well for my family and I until we found something bigger.
Cathay will pay you $14k if you are on a month-month rental, and you can pocket the difference if there is one. Once you sign a lease, Cathay will pay up to $22.8k (varies with the housing market) directly to the landlord (no pocketing the difference).
My suggestion would be to spend a few weeks in the Headland, ask around once you're here (check the bulletin board in the Cx crew area and online), move into a small furnished apt. temporarily in Tung Chung, and start looking for something more permanent once you're done the bulk of your training.
Good luck!