MD10 LNDG PHX EXPERIENCED A POSSIBLE LOSS OF SEPARATION DURING VECTORS FOR APCH.

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WE WERE ARRIVING FROM THE N (CAPT-PF) AND BEING VECTORED FOR RWY 7R. WHILE APCH WAS INSISTING ON A VISUAL APCH; WE REQUESTED (AT LEAST 5 TIMES) AN ILS (TO ANY RWY). WE WERE HDG 170 DEGS AND CROSSED THE RWY 7R FINAL AND WAS TURNED TO HDG 120 DEGS WHEN THERE WAS CONSIDERABLE CONFUSION BTWN VECTORS FOR OUR FLT AND OUR COMPANY FLT. THEY WERE FOLLOWING US; BUT SOMEHOW ENDED UP BELOW (4000 FT) AND BEHIND US. THEY WERE ISSUED A VISUAL (INITIALLY) TO FOLLOW A B737 TO RWY 7R. WE WERE THEN TURNED AND CLRED FOR THE ILS APCH; BUT BEFORE FULLY CAPTURED; WE SAW THE COMPANY ACFT AND ATC TURNED US BACK R TO HDG 120 DEGS. ATC INSTRUCTED COMPANY TO CLB TO 5000 FT; BUT THEN REALIZED WE WERE AT 5000 FT AND DSNDED THEM BACK TO 4000 FT. THERE WAS CONSIDERABLE CONFUSION AND RAPID FIRE CTLR INSTRUCTIONS COMBINED WITH APCHING DARKNESS AND RAIN DUE TO A BROKEN VARIABLE TO OVCST HIGHER LAYER. AT ABOUT 6 MI HDG 120 DEGS AT 5000 FT; THE CTLR ASKED IF WE SAW THE ARPT. WE INDICATED YES BUT TOLD THEM WE COULD NOT ACCEPT THE VISUAL BECAUSE OF OUR EXCESSIVE ALT. WE WERE THEN GIVEN A 360 DEG R TURN AT 5000 FT. WHEN ATC CONTINUED WITH COMMENTS ABOUT A VISUAL; I GOT ON THE RADIO AND REQUESTED A SHORT TURN ONTO THE ILS RWY 7R. WE WERE THEN GIVEN A VECTOR TO FINAL AND LANDED UNEVENTFULLY. I CALLED TRACON AND SPOKE WITH A SUPVR WHO TOLD ME THEY WERE LOOKING AT BOTH THE RADAR PLOTS AND VOICE RECORDINGS. THE SUPVR INDICATED THAT IT LOOKED LIKE EITHER OUR FLT OR THE OTHER COMPANY FLT ACCEPTED A CLRNC FOR THE OTHER ACFT. THERE WAS SO MUCH CONFUSION THAT A MISREAD CLRNC OR READBACK COULD HAVE OCCURRED. FROM THE FLT DECK; IT APPEARED THAT THE CTLR WAS CONFUSED AND ISSUED AN INCORRECT CLRNC FOR THE INCORRECT ACFT. POOR CTLR COORD AND SEQUENCING FROM OUR PERSPECTIVE. SEPARATION CONCERN: PHX TRACON WAS FAIRLY INSISTENT ABOUT ISSUING A VISUAL CLRNC AND SEEMED TO DISREGARD OUR REPEATED REQUESTS FOR AN ILS. WE WERE UNFAMILIAR AND WANTED THE PROTECTION OF THE ILS. THEY SEEMED BENT ON US GIVING A VISUAL. SUPPLEMENTAL INFO FROM ACN 605651: THE CTLR CLRED US FOR THE ILS RWY 7R APCH WHEN WE WERE WITHIN 1 NM OF THE FINAL APCH COURSE WHICH RESULTED IN US OVERSHOOTING FINAL TO THE S. AS THE CAPT TURNED THE AIRPLANE TO INTERCEPT THE RWY 7R FINAL APCH COURSE FROM THE S; WE RECEIVED A TCASII TA. I LOOKED AT THE TCASII DISPLAY WHICH SHOWED ANOTHER ACFT 700 FT BELOW US AND JUST TO OUR L. THE CTLR THEN GAVE US A HDG OF 120 DEGS (R TURN) OFF OF THE APCH. WE WERE THEN GIVEN A R 360 DEGS TURN AND OFFERED AN APCH CLRNC; BUT WE REFUSED THE CLRNC BECAUSE WE WERE TOO HIGH; TOO CLOSE TO THE ARPT. THE RADIOS WERE VERY BUSY AND TFC VOLUME WAS VERY HVY. I DON'T BELIEVE THERE WAS ANYTHING WE DID THAT CONTRIBUTED TO THE LOSS OF SEPARATION BTWN US AND THE OTHER ACFT.