The same thing happened with my iMac G3, after spending hours of work on it I found the problem was the RAM (strange, isn't it?), also I had to reset the PMU.
look here http://support.apple.com/kb/HT1431
Well, the battery may be not connected (it can't charga/it can't gove energy/it isn't recognised), my sister's iPod showed the same problem, the one thing you have to do is to give it back, it's most probably a logic board problem, impossible to fix.
Does the iPhone work well? Have you tried with more cables? with any other computer? Maybe it can't give 2A of power to the USB port, try disconnecting all and connecting only the iPhone. Let me know...
Wow, no chime, that's veeery bad! THat means the motherboard hadn't passed the self-check.
Have you done anything particular before the "death"? That's a problem that can't come out from nothing. If you want you can try an EPROM reset, by doing what is written here --> http://support.apple.com/kb/HT1379 Try even disconnecting the mac from power supply and keeping the power button held for some seconds (10-15)