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check out westsiderentals.com. your budget is just about right for west hollywood and west hollywood adjacent (east and south, not beverly hills, of course), so you should have no trouble finding a place.
areas to avoid depends upon your reason for wanting to avoid an area: safety; noise; traffic congestion; people congestion; public solicitation; other. the city of west hollywood is quite safe. the neighborhoods deteriorate from west to east, but i use the term VERY liberally, as most of west hollywood is quite desirable to your average city-dweller. pockets of western hollywood (sometimes also called "west hollywood") are not-- make sure you aren't misled by rental listings.
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