CREW OF C650 ENDURES MULTIPLE CLRNC; SPD; RWY CHANGES AT MDW. APCH CTLR APPARENTLY WAS TROUBLE MAINTAINING SITUATIONAL AWARENESS.

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I WAS THE PNF ON A FLT FROM NEWTON; IA; TO MDW; IL. WE WERE ASKED TO KEEP THE SPD UP TO 250 KTS; THEN TO REDUCE TO 180 KTS AND HOLD THAT TILL THE MARKER (ERMIN FOR ILS RWY 4R) AND THEN CONTACT THE TWR. A B727 (ACFT Z) WAS PUT IN FRONT OF US. THEY HAD WAY; WAY OVERSHOT THE RWY FINAL APCH. I CONTACTED APCH CTL AND ASKED IF HE WAS LNDG ON RWY 4R MDW AND THAT THE SPACING DIDN'T LOOK GOOD. HE ASSURED US THAT THE B727 WAS MUCH FASTER THAN US AND IT WOULDN'T BE A PROB. AT THE MARKER WE CONTACTED THE TWR AND REDUCED OUR AIRSPD. WE WERE CLRED TO LAND RWY 4R AND WERE ADVISED THERE WOULD BE AN ACFT XING OUR RWY 4R R TO L. THIS WAS THE ACFT WE HAD THE RA WITH; A B737 (ACFT Y). THE B737 WAS TAKING OFF RWY 31. ON FINAL; TWR TOLD US TO CHANGE OUR LNDG RWY TO RWY 4L; WHICH DOESN'T MEET OUR COMPANY OPS SPECS OF 5000 FT FOR TKOF AND LNDG. WE ADVISED TWR WE WERE UNABLE RWY 4L AND WAS RECLRED TO LAND RWY 4R. THE B727 LOOKED AS IF IT WAS HAVING NO PROB CLRING THE RWY BEFORE OR LNDG. THEN ON SHORT FINAL; TWR TOLD US TO GO AROUND USING THE WRONG N-NUMBER. I QUESTIONED THIS FOR 2 REASONS: 1) IT WASN'T OUR N-NUMBER; AND 2) THE B727 LOOKED LIKE IT WAS CLR OF THE RWY ONLY THE TAIL WAS POSSIBLY OVER THE HOLD SHORT LINE AT THE DEP END OF THE RWY. AND THERE WAS ENOUGH TIME FOR THE B727 TO WELL CLR BY OUR LNDG TIME. TWR THEN USED THE CORRECT N-NUMBER. TOLD US TO GO AROUND L TURN CLB TO 3000 FT. THE PF ASKED WHAT HDG. I TOLD HIM PASS BEHIND THE B737 AND POINTED THE ACFT OUT AND CLB TO 3000 FT. THEN THE TWR ASKED IF WE HAD THE B737 IN SIGHT. I DID; AND ACKNOWLEDGED IT. WE BOTH HAD THE B737 IN SIGHT AND WERE TURNING TO AVOID THE B737 AND DID SO WITHOUT ANY PROB. AS WE LEVELED AT 3000 FT; THE TCASII RA CAME ON 'MONITOR VERT SPD.' I HAD THE B737 IN SIGHT -- THINGS WERE JUST VERY TIGHT WITH B737 DEPARTING RWY 13 WITH A NE TURN; AND US GOING AROUND RWY 4R WITH A RELAXED TURN. I TOLD THE TWR WE HAD A TA AND CORRECTED THAT TO AN RA. THEY SAID TO MAINTAIN VISUAL SEPARATION; WHICH WE DID. THE B737 THEN GOT ON THE RADIO AND SAID THEY JUST HAD TO TURN TO AVOID US; AND THAT WE PROBABLY NEVER EVEN SAW THEM. TWR TOLD B737 THAT WE HAD A VISUAL ON THEM. I DON'T THINK THE TWR LIKED THAT WE DIDN'T ACCEPT RWY 4L. THE B737 NEVER HAD TO MAKE ANY TURN AND WE HAD THEM IN SIGHT ALL THE TIME. DEPARTING THE B737 WITH NE TURN WAS NOT A GOOD IDEA -- IT LEFT US VERY LITTLE OPTIONS IF WE WERE TO GO AROUND. B727 COULD HAVE CLRED RWY QUICKER OR BE PUT BEHIND US ON FINAL. WE COULD HAVE SLOWED SOONER ON RWY 4R APCH. BUT WHAT ABOUT ACFT BEHIND US?