I have lived in Westwood for most of my life (I'm 50). I own an older garden apartment building in Westwood just off of Santa Monica Blvd.. I have had Australian tenants, previously (some with no American credit rating). I take a credit report all the time, these days, though. That's something most landlords require before making a decision to rent. I can turn you down just for having no credit history so that may be a problem for you. It is pretty much a landlords market right now, too.
I can't believe anyone would suggest the San Fernando Valley - especially Woodland Hills - to anyone at this time, if they are going to be working in the Beverly Hills area. The traffic on the 405 is MURDER with the construction that is going to go on for probably up to another 2 years (not to mention all the gas and wear and tear on a vehicle). This the most expensive road construction project ever in America, already. My cashier at Ralphs on Olympic & Barrington finally got a transfer to near his home in Woodland Hills. He use to spend up to 90 minutes ONE WAY in traffic.
You shouldn't have to pay for water. Most buildings pay for it. I agree that gas is cheap. Gas heating and gas hot water heating is a big plus. I pay about $12 a month in the summer. The most I've ever paid is $75 for one month in the winter. LADWP (Los Angeles City) power is cheaper than any other city. Lowest bill you can have is $72.15 every 2 months (LADWP bills every 2 months). That includes a nearly $50 Sanitation (Trash) fee. My normal bill is about $150 every two months.
I have a 2005 Acura TL. I pay about $2500 per year car insurance. I have much more than the normal minimums but no tickets.
If you can learn how to play the restaurant coupon game, eating out in Los Angeles is actually VERY cheap. There is an incredible amount of competition here.
Your best bet is the Westside, period. It never gets hot so you'll save on a/c bills and all the fill-ups for driving further away not to mention the time. Plus there is plenty to do all the time on the Westside. Nobody here has mentioned Craigslist for apartment hunting
http://la.craigslist.org/apa/ . Yes, you have to watch out for some bogus Craigslist listings but I advertise on there. It will give you, at least, an idea of what rentals are near your work.
You can get an older but cute 2 bedroom, 1 bath place on the Westwood for around $2K (slightly cheaper in some Westside areas). I should know, that's what I rent my apartments for. Hardwood floors and all...