I am not an expert on the FI system.
There are a few different manuals I think do a pretty good job of explaining its operation. Here are a couple links.
http://www.njfiats.org/images/stories/s ... manual.pdf
http://www.mirafiori.com/faq/fiatFI_may2002.pdf
Yes the ECU does need a signal from the ignition to time the injectors. The coil should have 2 brown/white wires on the "D" terminal. One is for the Tachometer the other is for the ECU. All the injectors fire at the same time. Interestingly the ECU grounds and the Temperature sensor ground wires make their grounds with two of the bolts holding the throttle body to the intake manifold. The ECU + voltage and the injector + voltage are supplied through the dual relay. So if the relay is good when the fuel pump is operating the ECU and the injectors should all have voltage to operate.
The fuel injectors can get stuck from sitting. Although its unlikely all 4 would have this happen have you applied voltage and ground to each injector to see if they click?