I've always had a hard time getting into download mode, and my phone works fine. I just spent 5 minutes trying different combinations until I got it.
I've heard people say trackpad-power button, volume down-home-power button, volume up/down-put battery in-plug in USB, etc etc etc. But the only method that works for me is to put in the battery, plug in the USB, wait for the charging screen, then hold volume down-trackpad-power, releasing power as soon as the screen goes black
You say your phone is turning on as soon as the battery is inserted? What if you have it plugged into USB first?