Greetings fellow Illinois M45 owner!
I had this same problem shortly after I got my car, it is called a resonator. Mine rusted and broke in the same exact spot, and I just had my local muffler shop weld in a straight pipe in it's place simply because I needed the car immediately, and he did not have the part in stock. The resonator is supposed to minimize or remove certain harmonics at different RPM's from what I understand. The only difference I notice is around 70 mph on the highway (and at 2-thousand-something rpm) I will hear a hum. Not real loud, but not where I would want to set my cruise for a long period of time.
I have a feeling that this may have been just about broke on your car for a while if you had a rumble on start-up. The car is supposed to sound quiet. Here's a link to what mine sounded like the night before I got it fixed:
http://www.youtube.com/watch?v=VRa_vSSSiLQ
Now it's almost as quiet as a Lexus. I wish it had more of a sound to it, but after thinking about re-doing the entire exhaust, I wasn't sure if I wanted it to sound like a muscle car all the time, and I didn't want to mess with possibly reducing the HP by not having enough back pressure.
My car came from TX, so I didn't think it should have rusted and broke after only 3 years, but the nearest Infiniti dealer in Barrington, IL would not cover it under warranty (because the heat shield on one of the cat converters had a rock-chip on it which they thought might have caused the resonator to "break")and wanted over a grand to replace the entire cat system (because it is supposedly all 1 piece)
Either way, keep us posted if you do happen to re-do the entire exhaust, I would be interested in how it sounds!