2007 Toyota Prius multiple issues

The transmission of Prius is not an automatic transmission. It is part of the hybrid system, having a planetary gearbox and two electric motors in it. It is a good guess, though: a bad transmission certainly could cause the issues you're seeing.

It could also be anything in the hybrid system such as a power inverter for any of the motors, or some control system, or even a battery that doesn't have enough power to propel the car, just to start the engine.

I would proceed by reading the error codes. These hybrid systems are purely electronically controlled, so chances are the car already knows what's at fault.

You could also try these simple steps to obtain more information:

• How much charge does the HV battery have? There should be a display in Prius.
• If the HV battery is nearly empty, what happens if you let the car "idle" (=recharge) with the engine on? Does the engine temperature reach operating temperature with the HV battery charge increasing?

Don't try changing transmission fluid as there is none in Prius.