A B737-800 FLC OVERSHOOTS THEIR ASSIGNED ALT 5 NM NW OF RIC; VA.

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DEPARTED RWY 2 CLBING TO 3000 FT MSL. FO FLYING ACFT WITH AUTOTHROTTLES INOP AND AUTOPLT OFF. MADE SEVERAL HDG CHANGES IMMEDIATELY AFTER TKOF DUE TO TSTMS NW OF FIELD. CAPT INVOLVED WITH COMMUNICATING TO ATC; AVOIDING WX; MANUAL SETTING OF THROTTLES; AFTER TKOF CHKLIST AND PERCEIVED FMS PROB. RESULT WAS CAPT MISSED 1000 FT BELOW CALL AT 2000 FT MSL. FO CLBED TO 3200 FT MSL AND RECEIVED AURAL ALT WARNING; ACFT THEN CLBED TO 3400 FT MSL BEFORE INITIATING DSCNT TO 3000 FT MSL. NO ACKNOWLEDGEMENT OF ALTDEV MADE BY ATC. CONTRIBUTING FACTORS: AUTOTHROTTLES INOP (CAPT MADE MANUAL ADJUSTMENTS. DEVIATING AROUND TSTMS. NO AURAL ALERT FOR ALT PRIOR TO REACHING ON B737-800. FMS WAS NOT PROGRAMMED TO ADJUST CLB SPD OR WAS SLOW TO CHANGE. LOW ALT LEVELOFF IN A BUSY ENVIRONMENT. 2 EVENTS COULD HAVE CHANGED THIS OUTCOME. 1) 2 MAN ACFT NOT DISPATCHED WITHOUT AUTOTHROTTLES. 2) LOW ALT LEVELOFFS CREATE COMPRESSION OF ACFT DUTIES AND IF AIRSPACE PERMITS; HIGHER ALTS SHOULD BE ASSIGNED. SUPPLEMENTAL INFO FROM ACN 483268: TWR GAVE US 2 HDG CHANGES TO GET ON COURSE AND AVOID TSTMS NEAR THE FIELD. TWR TURNED US OVER TO DEP CTL WHO GAVE US 5000 FT MSL WITHOUT COMMENT. A CONTRIBUTING FACTOR IS THAT OUR B737-800'S DO NOT HAVE A 1000 FT TO GO AURAL WARNING -- ONLY A 300 FT ALTDEV WARNING. I BELIEVE A 1000 FT TO GO AURAL WARNING WOULD HAVE ALERTED BOTH THE CAPT AND I TO THE RAPIDLY APCHING ALT RESTR AS WE WORKED ON AVOIDING THE WX AND FOLLOWING DEP INSTRUCTIONS.