B737-200 CREW MADE MULTIPLE GAR AT DAL.

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GOT HIGH ON VISUAL APCH TO RWY 13R AT DAL. DIDN'T CONFIGURE EARLY ENOUGH; APPARENT STRONG TAILWIND ALOFT (OVER DFW) MAGNIFIED PROB (-200 ACFT; NO CDU WINDS). VISIBILITY TRICKY DUE TO SUN ANGLE IN MORNING; CAUSING DISTR IN LOOKING FOR GOOD GND REFS. HAD RWY 13R IN SIGHT OK; BUT MARGINALLY. WX RPTED AS 6 MI; BROKEN; BUT WITH SUN ANGLE SEEMED MORE LIKE 3 MI AND HAZE. TOO MUCH ENERGY TO DISSIPATE; (170 KIAS; FLAPS 30 DEGS; NOT COMING DOWN LIKE WE WANTED). CALLED FOR GAR. EXECUTED OK; STAYED WITH TWR; L CLOSED TFC FOR RWY 13R AT 2000 FT MSL. SECOND TIME AROUND; COULD SEE FIELD VERY EASILY ON DOWNWIND; BUT VISIBILITY DETERIORATED AS WE PROGRESSED THROUGH BASE AND FINAL TURN DUE TO SUN ANGLE/HAZE. DURING BASE TO FINAL; GOT TCASII TA/RA FOR HELI TFC (RPTED BY TWR) BELOW US. MANY DISTRACTIONS AND VERBALIZING CHKLIST; TALKING TO TWR ABOUT TFC AND LNDG CLRNC; TALKING AMONGST OURSELVES ABOUT RESOLVING TFC CONFLICT; TRYING TO KEEP FIELD VISUALLY IN SIGHT; TCASII SYS SQUAWKING AT US. BEFORE WE KNEW IT; WE HAD MISTAKENLY LINED UP ON FINAL FOR RWY 13L. TWR AND WE CAUGHT THE ERROR SIMULTANEOUSLY; (AT 1000 FT); BUT TOO LATE TO SWITCH OVER TO RWY 13R. LEARJET ON AND HOLD RWY 13L TOLD TO EXIT RWY QUICKLY; NOT QUICK ENOUGH; SENT US AROUND ON SHORT FINAL AS LEAR FINALLY CLRED RWY (LAG PURSUIT). MADE ANOTHER GO AROUND; THIS TIME L CLOSED TFC FOR RWY 13R AT 3000 FT. MUCH BETTER VISIBILITY OF EVERYTHING UP HIGHER. WENT TO MIX MASTER AT 3000 FT. NORMAL VISUAL AND LNDG ON RWY 13R FROM THERE. BY THE WAY; I MADE PA'S TO PAX BOTH TIMES AND KEPT THEM INFORMED. 2 DEPLANING PAX MADE POSITIVE COMMENTS ABOUT OUR SAFETY AND KEEPING THEM BRIEFED; THEY APPRECIATED IT. NEXT TIME ASKING FOR FULL ILS APCH DURING THOSE 'TRICKY' VISIBILITY CONDITIONS IN MORNINGS APCHING INTO SUN; EVEN THOUGH ATIS WX LEADS YOU INTO THINKING IT IS OK. ALSO SOMEWHAT HANDICAPPED IN -200 NOT KNOWING THOSE WESTERLY WINDS ALOFT OVER RANGER VOR/DFW. DISPATCH WX WINDS ALOFT PAGE (ON RELEASE PAPERS) WAS MISSING/OBLITERATED. APOLOGIES TO TWR; APCH; ETC. WENT OK. SUPPLEMENTAL INFO FROM ACN 538429: ATTEMPTED TO MEET THE NORMAL GS AT FLAPS 30 DEGS AND APPROX 160 KTS; BUT UNABLE AND EXECUTED A GAR. WHILE AGGRESSIVELY LOOKING FOR THE TFC; COMPLYING WITH THE TCASII TA/RA; AND RUNNING THE FINAL DSCNT CHKLIST; AND LOOKING FOR THE RWY ACROSS COCKPIT; I MISTAKENLY ALIGNED THE ACFT WITH RWY 13L; INSTEAD OF RWY 13R. I WAS FLAPS 5 DEGS OVER DFW AND CONFIGURED TO FINAL APCH CONFIGN SHORTLY THEREAFTER; UNFORTUNATELY; I DID NOT RECOGNIZE AND COMPENSATE IN TIME FOR THE TAILWIND CONDITION. ON THE NEXT APCH; I PROBABLY SHOULD HAVE INSISTED ON THE FULL ILS DUE TO THE REDUCED VISUAL LOOKING INTO THE SUN. ON THE SECOND APCH I ALSO ALLOWED THE TFC/TCASII TO DISTRACT ME; COMBINED WITH THE MARGINAL VISUAL AND MADE THE MISTAKE OF LINING UP ON THE EASIER TO SEE RWY. I SHOULD HAVE XCHKED THE LOC MORE; BUT ALLOWED THE EXTERNAL FACTORS TO TAKE PRECEDENT. AS WE APCHED THE RWY 13L CTRLINE IT BECAME EASY TO SEE AND I DIDN'T RECOGNIZE MY MISTAKE IN TIME.