B757-200 CREW WAS GIVEN MULTIPLE STARS AND RWYS ARRIVING INTO LAX. THE CREW HAD A TCASII RA AFTER ANOTHER ACFT WAS PUT AHEAD OF THEM.

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INBOUND LAX CLRED CIVET. BRIEFED AND SET UP FOR APCH TO RWY 24R. TOLD TO MAINTAIN 310 KTS. CLRED TO DSND ON CIVET PROFILE. STARTED DOWN AND REBRIEFED FOR RWY 25L. IN DSCNT; FREQ TOO BUSY TO GET A WORD IN WITH CTLR FOR SLOWER. SLOW ANYWAY GOING THROUGH 12000 FT. NEXT; TOLD TO TURN R 10 DES FOR RWY 24R AND MAINTAIN 8000 FT. NOW W OF DENAY STILL HIGH AND FAST. RETURNED AND BRIEFED FOR RWY 24R. TOLD TO MAINTAIN 5000 FT AND LOOK FOR TFC TO FOLLOW TO RWY 24R. STILL HIGH AND FAST. FREQ TOO BUSY TO GET LOWER/SLOWER REQUEST IN. CLOSING ON TFC AHEAD RAPIDLY. ASKED BY TWR IF RWY 24L IN SIGHT. YES. GIVEN CLRED TO 3000 FT; CLRED VISUAL RWY 24L; MAINTAIN 180 KTS TO FAF. STILL HIGH AND FAST. DROPPING GEAR AND FLAPS EARLY TO TRY AND SLOW CLOSING ON TFC RAPIDLY. PASSED THROUGH 1000 FT AGL. UNSTABILIZED; TOO FAST. STARTED PLANNING WITH PNF FOR GAR. INITIATED GAR AND AS ACFT ARRESTED DSCNT; GOT AN RA FOR 'CLB' -- IMMEDIATELY FOLLOWED BY 'CLR OF CONFLICT' AS PARALLEL ACFT CONTINUED DOWN HIS PATH AND WE CLBED OUT. CONTRIBUTING FACTORS: MULTIPLE RWY CHANGES. MULTIPLE FREQ CHANGES TO BUSY FREQS. ACFT STEPPING ON ONE ANOTHER'S XMISSIONS. CTLRS TOO BUSY. KEPT TOO HIGH; TOO FAST FOR TOO LONG BY ATC. ACFT VECTORED IN FRONT OF US ON APCH AT A MUCH SLOWER RELATIVE SPD CAUSING LAST MIN RWY CHANGE.