In short, find a better driving instructor and, if possible, learn to drive a car with manual transmission.
The long (still simplified) answer:
Does the accelerator control and ensure gas is flowing through the engine? The harder I press, the more gas flows in and the faster is the engine RPM. Is my mental model correct? Also, if this is the case, does it mean unless I step on the accelerator, no gas flows in and the engine does not spin (unless it was spinning, because of inertia)?
Generally, yes, but there are other factors that affect the amount of fuel allowed in the engine:
- Idle. When the engine is turned on, some fuel is allowed in the engine even with the accelerator pedal fully released. This is why automatic cars inch forward when the transmission is in "D" position. You need to apply brakes to prevent this. This is also how e.g. steering assistance or air conditioning are powered.
- The actual amount of fuel depends also on the engine RPM at the moment.
- Other, less important factors also impact the engine fuel consumption/power.
While driving at, suppose 50kmph, suppose I take my foot off the accelerator, and neither do I press the brake. Does it mean the engine will still spin for sometime (because of inertia) and the car will move some distance (may be even a few miles) before eventually coming to a halt? Is it like when I stop pedaling when riding a bicycle?
Mostly. An automatic car will not, in fact, completely come to a halt on a level road - see about the "idle mode" above.
Slopes can alter this behavior.
Uphill, the idle engine power may not be enough to creep the car forward.
Downhill, the car can gradually accelerate all the way to its designed maximum speed (and even more) with no action on the accelerator pedal. It is up to the driver to apply brakes and common sense.
What happens if I press both the accelerator and foot brake together? As one tries to move the car, and another resists, will some parts internally grind against each other, eventually damaging the parts, maybe even an accident? Do not want to try this, just want to know what happens in that case. Or is the car somehow smart enough to not let the engine spin when the foot brake is pressed?
In a sane car, brakes outpower the powertrain by a great margin. If you press on brakes strong enough, no pressure on the accelerator pedal will make the car move.
Don't really try this, a wrong order of releasing the pedals can make you car vigorously jump forward, up to and including crashing into some obstacle.
Yes, if you insist on pressing and holding the accelerator and the brakes together, some parts wear intensively and can overheat in a few minutes. Repairs will be expensive.
This (among other things) is why automatic cars are controlled by a single foot and this is one of the first things you learn in the driving lessons.
Then again, pressing both pedals (to an extent) is sometimes used when driving off-road.
Does pressing the foot brake, in automatic, also decouple the transmission (wheels) from the engine, in addition to resist the wheel with brake shoes? Is it similar to neutral in manual?
Not really. An automatic transmission always applies some torque to the wheels. If the accelerator is not pressed, the internal friction that applies this basic torque is considered safe. You are not expected to manually disengage the transmission when waiting for a green light.
If you really want to disconnect the transmission, there is "N" position on most automatic transmissions. "P" is the same, but it also applies an internal brake mechanism that blocks the wheels.