No the TPS is only a trim sensor... the car can be driven without it, throttle response and idle just suck and it'll like to stall if the IDL switch is damage or the TPS is actually disconnected.
Yes, the injectors will have power with the ignition in the ON position... they won't have a ground as that's supplied by the ECU.
Engine light??? You replaced the check engine light or what? Pull the glove box out and test the ECU for +12v at Batt, B+ and B1+. I can't provide a pinout as you haven't filled out your profile or mentioned in this thread what year your car is... but those are the pins you need to check. Batt should always have power and +B and +B1 are both ignition switched.
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