B737 Captain reports a late descent clearance from Approach Control in IMC with a strong quartering tailwind. The approach becomes unstabilized and the crew elects to go around without pushing the TOGA button. This requires the pilot flying to ignore the flight director and attempt to fly raw data missed approach causing minor altitude deviations.

Narrative

Flight was nearly 2 hours late due to a scheduled late report. Captain had 8 hours minimum rest due to dead heading delays the day before. First Officer indicated he himself didn't sleep well the night before. On approach; weather went below forecast with moderate rain. Airport was landing with tailwind landing components nearly at the limits. Aloft; there was a quartering tailwind of approximately 20 KTS. On base leg; we were held high due to crossing traffic underneath. This put us well above GS outside of the marker. Approach offered to give us another box pattern because 'we didn't start down right away' (we were configuring) but we were high due to proximity traffic which delayed clearance. Nonetheless we felt we could recover within the IMC stabilized approach criteria; so we continued. However; on the assigned heading the crosswind pushed us north of the localizer and most likely the FD didn't fully center; so it never coupled. This lead to a full-scale deflection at about 1;200 FT AGL. We elected to execute the missed approach and Tower gave us a turn to 190 and a climb to 2;000 FT. During the missed approach the FD kept attempting to descend us to 1;500 FT (the initial approach altitude.) This forced us to fly raw data while we cleared the FD so we could re-arm the system from scratch. There were minor altitude deviations while we conducted the missed raw data.At first we should have requested another runway. After landing; we saw departing aircraft already taxiing for the opposite runway. Then when we were kept high for crossing traffic; we shouldn't have attempted to 'salvage' a busted approach. During the go-around it's possible the TOGA switch didn't get pressed; which may have been a cause of the erroneous FD commands. All of the above wasn't helped by the fact I was on minimum rest.