B737 CREW; ON FINAL TO RWY 4R AT ORD; REALIZED THEY DIDN'T HAVE LNDG CLRNC AND EXECUTED A GAR; PUTTING THEM IN HARM'S WAY WITH DEPS ON RWY 9L.

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BEGINNING APCH TO RWY 4R; APCH WAS ARMED BUT UNSTABLE. HAD RWY 4R IN VISUAL. CONTINUED APCH AT REKKS; WAS ABLE TO STABILIZE. STILL HAD RWY 4R IN SIGHT BUT MISSED TWR HDOF UPON REALIZING NO LNDG CLRNC. WENT MISSED IN THE GAR. NOTICED ACFT IN DIRECT CONFLICT HEADING TOWARD US. MAINTAINED VISUAL WITH RA ACFT UNTIL RE-ESTABLISHING CONTACT WITH ATC. DID ANOTHER APCH TO LNDG ON RWY 4R ORD. SUPPLEMENTAL INFO FROM ACN 542727: MISSED APCH RWY 4R ORD DUE TO LACK OF LNDG CLRNC. ON MISSED APCH; HAD VISUAL CONTACT WITH ACR Y. ACFT APPARENTLY DEPARTING RWY 9L. STARTED PUBLISHED MISSED APCH; BUT THEN HAD TO DEVIATE TO AVOID COLLISION WITH ACR Y. SUPPLEMENTAL INFO FROM ACN 543281: ON TKOF FROM RWY 9L AT ORD; WE HEARD NUMEROUS BLOCKED AND GARBLED XMISSIONS FROM TWR. AT ABOUT 1500 FT MSL TWR ADVISED US OF AN ACFT AT 2 O'CLOCK POS; 1400 FT ON A GAR OFF RWY 4R THAT WASN'T ON THEIR FREQ. LOOKING IN THAT DIRECTION; I SAW ACR X; A B737; ABOUT 1 MI AWAY; CONVERGING WITH US IN A SLOW R TURN TO THE E; AND ABOUT 200 FT BELOW OUR ALT. I POINTED OUT THE TFC TO THE PF; AND HE BEGAN TO TRY TO OUTCLB IT. AS I WAS ACKNOWLEDGING THE TFC TO ATC; WE GOT A TCASII MONITOR VERT SPD RA. THE PF LEVELED OFF TO COMPLY WITH THE RA. AS I WAS ANNOUNCING THE RA TO ATC; WE GOT A DSND RA AT 2400 FT. AS WE STARTED DSNDING; ATC APPROVED A L TURN AWAY FROM THE TFC. AT THIS TIME WE GOT AN OVERSPD WARNING AS THE AUTOTHROTTLES WERE AT CLB THRUST AND WE WERE STILL CONFIGURED FOR TKOF. CONCURRENTLY; WE GOT A GPWS 'DON'T SINK' WARNING. I TOOK MY EYES OFF THE TFC LONG ENOUGH TO CONFIRM THE FLAP OVERSPD AND ACFT ENERGY STATE; AND RECONFIGURE THE FLAPS. BY THIS TIME THE B737 HAD COMPLETED ITS TURN TO THE E AND WAS OUT OF SIGHT IN THE CLOUDS BUT SHOWING 400 FT ABOVE US AT .5 MI ON THE TCASII ON A DIVERGENT COURSE.