B737-700 FO ON APCH TO SAT FOLLOWS UNRELIABLE GS AND GETS LOW ALTITUDE WARNING FROM CAPT AND TOWER.

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WE WERE ON AN ASSIGNED HDG TO INTERCEPT THE APCH COURSE JUST OUTSIDE OF HASDO AT 3100 FT. WE WERE CLRED FOR THE APCH ILS RWY 12R. I SELECTED VOR/LOC INITIALLY THEN SELECTED APP AS THE ACFT (ON AUTOPLT) INTERCEPTED THE LOC. I CONFIRMED LOC AND GS CAPTURED ON THE FLT DIRECTOR ANNUNCIATOR PANEL. AT APPROX 2200 FT MSL; I HEARD THE CAPT/PM STATE MY NAME AND SAY; 'WE ARE LOW; START A CLB BACK UP TO 2800 FT.' AS I WAS CLBING BACK UP TO 2800 FT I HEARD 'CAUTION OBSTACLE' AND SAY SAME ANNUNCIATION. THREATS AND ERRORS 1) DID NOT KNOW GS WAS RPTED UNUSABLE ON THE ACARS WX RPT. I HEAR TWR CLR ANOTHER ACFT FOR THE ILS GS UNUSABLE APCH; BUT DIDN'T VERBALIZE IT AND THEN FORGOT ABOUT IT WHEN I WAS CLRED FOR THE APCH. 2) WX WAS MARGINAL VMC SO I BRIEFED ILS RWY 12R AND THE VISUAL APCH PLANNING ON A VISUAL APCH. 3) I VERBALIZED VOR/LOC SELECTED BUT NOT APP SELECTED. 4) WHEN I CONFIRMED LOC AND GS WERE CAPTURED; I APPARENTLY BELIEVED I WAS ON THE PROPER GS AND HAD TUNNEL VISION ON THAT EVEN THOUGH AFTER THE FACT I REALIZED I HAD INTERCEPTED A FALSE GS. I STILL CANNOT BELIEVE I MADE SUCH A STUPID MISTAKE. 5) ATC DID NOT VERBALIZE 'GS UNUSABLE' TO US WHEN CLRED FOR THE APCH. 6) THE PM TOLD ME AFTER WE WERE ON THE GND THAT THE TWR CTLR TOLD US TWICE TO CLB BACK UP TO 2800 FT. I NEVER HEARD ANY OF THOSE CALLS. WHY I DIDN'T HEAR THEM; I DO NOT KNOW. FOLLOW OUR PROCS ALL THE TIME.