Hola Expat -
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I have no recent news about the conditions with KAL. Some 15 years ago, many friends (former PanAm colleagues) had joined KAL flying 747-200 and were based in the US, most were Anchorage base (cargo), and some were based on the West Coast (LAX/SFO). I am gone from the US since that time and lost contact with them. I remember them being quite satisfied with their rosters, and ample time to commute from their US home, sometimes on the US East Coast. The one complaint was that their F/Os and F/Es did converse in Korean, very poor English... few F/O ever were sufficiently trained to handle a takeoff or landing, so they said that they "soloed" the airplanes.
xxx
As far as buying a house in the US, no problems there for any foreigner to buy a house. The housing market is severely depressed right now in the US, and the US $ exchange rate is at a record low, unfortunately, your KAL contract will be probably in dollars... As far as buying , there are bargains.
xxx
Since you will enter and exit constantly on a crew visa, no problems, anyone can buy a house (call it a "second residence" as opposed to your European domicile). However, for your family to be there full time, will require them to hold a residence visa, as they would be limited to 90 days "visits" with their European passport. Get advice from an immigration lawyer there. Watch-out about your status also for income tax if you are considered to live there by the IRS (Internal Revenue Service, the tax people).
xxx
I used to live in the LA area when I lived in the US, I recommend the areas known as "Orange County" and "Ventura County" as the best for a family residence. Closer to LAX, the areas North of the airport are acceptable, that would be "Santa Monica", and the West side of the "San Fernando Valley". And first, you will need a car, public transport in LA is worthless. Cars, as you probably know, are inexpensive. You can buy a new/used car in two hours' time there, including insurance and registration. Learn the freeways to drive around on the maps. Everything is "many miles away"... from where you are. LAX crew parking is non-existant (for KAL and others), so, park your car at the hotel used by KAL crews (in the Santa Monica area if not changed) and ride the hotel bus to the international terminal.
xxx
Happy contrails