When the idle drops, does your radio volume go down? Lights dim? Gauges dim?
If yes, and taking into consideration that your batt and alt tested good:
It could be grounding out somewhere serious (this would put extra load on batt and alt). Only place I know of that has wiring big enough to completely drain the system (which would give you this problem) would be the connection to the starter.
Possibly coincidental that it stayed running with the jumper cables, if this is the case, it may be the IAC.