A320 CREW HAD ADVERSE WX AND TSTMS ON APCH TO MMMX.

Narrative

ROUTINE FLT TO MEX. SET UP FOR RWY 5R ILS AND AS WE CROSSED SMO; ATC DIRECTED US TO FLY DIRECT MEX VOR; RWY CHANGE IN PROGRESS. ABOUT 5-6 MI NE OF MEX VOR A TSTM WAS MOVING SW TOWARD ARPT. ATC KEPT US TOO HIGH WHEN THEY TURNED US IN FOR RWY 23 AND WE ELECTED TO GO AROUND. ATC REVECTORED US TO RWY 23 AND NOW TSTM APPROX 4 MI NE OF ARPT. WE CHEATED JUST A TAD ON THE VECTOR TO AVOID TSTM (HVY RAIN AND SOME HAIL) AND ENCOUNTERED 30-45 KT TAILWIND WITH MODERATE TURB AS WE TURNED FINAL AND ELECTED TO CONTINUE DSCNT TO STAY VFR; BUT LEFT GEAR UP WITH INTENTION OF GOING AROUND AGAIN AND GETTING AWAY FROM TSTM. WE DSNDED TO APPROX 700 FT AGL AND GPWS; TOO LOW GEAR TRIGGERED. WE OVERFLEW ARPT ON MISSED AND ATC TOLD US THAT ANOTHER RWY CHANGE WAS IN PROGRESS. WE WERE VECTORED BACK TO SMO AND SET UP FOR RWY 5R ILS AND LANDED WITH TSTM NOW OVER DEP END OF RWY. AS WE SLOWED THROUGH APPROX 80 KTS; THE WINDSHEAR ALERT SOUNDED. AS WE TAXIED TO GATE; TSTM CAME OVERHEAD WITH HVY RAIN AND LIGHT HAIL AND APPROX 40-45 KT GUSTS. INTERESTING DAY. BEING VERY FAMILIAR WITH MEX WAS A HELP. EVERYTHING WAS REASONABLY SMOOTH AND RETHINKING THE SIT I WOULD NOT HAVE DONE ANYTHING DIFFERENTLY.