Say that my electric motor battery is totally dead, can the car still run with just the gasoline motor?
The only known way of depleting the battery-pack to that extreme is running out of gas, yet continuing to drive anyway. Prius goes out of its way from preventing deep discharges. 1 bar actually means you've got over 40% battery left still. So, short answer, no. .
If you physically removed the motor (MG2), no. if you removed the battery, no. if the battery was totally flat, maybe. Provided there is enough power to start the ICE then it will move, and MG1 will start charging the HV battery in a major panic mode. It would also be a bit of a crappy drive, since you only have 90something horsepower and bugger all torque.
Hi TM, If the motor failure is a short to ground. Which is sensed as a ground-fault (just like droping a hair-dryer in a sink of water and why we have GFI plugs in bathrooms). With a ground-fault, the computers will not power up the motor, or the hybrid synergy drive, and as there is no toqure from the motor to the transmission except as biased by the counter torque provided by the hybrid synergy drive motors, engine torque, even if the engine is running, can't make it to the wheels. A shorted winding (without a ground fault) allows the cars to still run, but it makes a buzzing sound, as the shorted winding is selected, and provides no motive power. People have driven their cars like this, but not long. This is the usual Gen 1 Transmission failure, believed to be caused by extra metal dust from the power sharing device (PSD) sleeve bearings. The Gen 2 and 3 Prius use needle (roller) bearings in the PSD, and thus have much less metal debri in the transmission.
No you need enough battery power to control the two electric motors, or the ICE would just spin and if there is not resistance to the electric motors the ICE can't power the front wheels.
The two electric motors work together with the planetary gears to "be" the transmission. Some engine power goes straight through the gears. Sometimes, power is pulled off from MG1 and is fed back to MG2, sometimes the other way around. Without either motor, this scheme wouldn't work. Theoretically, the system could work with no battery, but it's not programmed to. Of course, with no hybrid battery, the engine can't even start.
However, it can (and will) work with very minimal power supplied to or from the battery - for example, if the battery is severely over temperature, as it might be on a very hot day after a lot of city driving with fast acceleration and stops, then the engine will run all the time, and the current to and from the battery is minimized to prevent additional heat from being generated. I believe a taxi driver noticed this happening once, for example.
It happened to me in summer (so I could be the driver you refer to). It was hot (about 30c) and humid but also a busy day so the car got non stop hard driving for about 8 hours, at which point the engine ran all the time and the car really tried to not use the electric motor. It was almost like the cold start running proceedure but the car was really revving compared to normal, probably to power the a/c. I thought I'd broken it!
I think there is a way to tie a large rubber band onto the engine and accomplish this direct drive. You can also mow the lawn with your Prius in this mode according to my relatives who say where do I wind it up.