Need some more details. I presume the battery was dead when you jump started the car the first time (from the engine relay box)? When you did this was the car battery still connected up front and presumably that battery was dead, otherwise why would you be jumpstarting?
Some observations:
- if the battery was dead you have erased all of your fuel trims in the ECU. If you have other issues with the engine which are causing you to run with a lot of fuel trim then all of a sudden having 0 trim may result in erratic / poor engine operation
- if the car battery was really dead and accepting a very high charge current the alternator may be having difficulty establishing normal voltage. If the voltage was really erratic it may have created some out of range errors on the ECU; however, the actual ECU is quite tolerant of lousy voltage. What is not tolerant of really low voltages are the ignition coils and the fuel injectors. If during the initial jump the car voltage was really low the spark will be weak because of insufficient dwell current and the injectors are slow to open resulting in less fuel delivery. Most ECUs attempt to adjust the calculated injector PW and the dwell time to adjust for voltage variations; but, those correction factors probably don't go below 10 volts. If your voltage dropped below the lower limit for the corrections you would have weak ignition and low fuel delivery. Lean fuel mixes typically result in surging at idle which was probably complicated by the DBW trying to control the idle.
- which dash lights were flashing? If the surging RPM was bad enough this may have triggered the misfire detection circuit in the ECU. Under persistent misfire the MIL will start flashing to indicate a misfire condition.
When you restarted the car the second time, the connection of the jump pack may not be relevant. Chances are that you put some charge on the car battery during the first start which made it easier for the alternator to stabilize the voltage eliminating the erratic engine operation during the second start attempt. That is my best educated guess without more specific details.
The other possibility is that when you jumped the car from the terminals at the engine relay box, you had a bad ground connection. This would contribute to erratic voltages and operating problems. However, if the starter motor spun over quickly during the first start attempt then the connection of the jump pack was likely not the problem.
