However, with a manual transmission, taking your foot off the accelerator induces engine breaking, so while you might use less fuel, the car will slow to a stop far more quickly.
Yes, that is true. I'm not sure what uses more gas, coasting in neutral or coasting in gear and blipping the throttle as needed (any difference is probably negligible) but coasting in neutral is still dangerous and shouldn't be done.
But engine braking can be good though. For example, there's a hill near my house, speed limit is 40. When I'm going down it, I'll start at the top at 40 in 4th, and just coast to the bottom with no brakes, still in gear, and my speed might increase to about 42 or 43. A quick tap of the brakes at the bottom, and I'm back down to 40. I'm putting much less wear on my brakes than the guy who rides the brakes all the way down. So, depending on the roads and traffic, the increased engine braking can be a good thing.
If a car cuts off the fuel, that is called a stall.
A car cannot run without fuel.
You are talking about idle. Which uses less gas to run.
A stall is not cutting off fuel. A stall is when the engine speed slows down enough that the engine can't move the car, and it stalls. In my above example, when I go down that hill in gear and coasting, my car doesn't use a drop of fuel. The movement of my wheels are enough to keep the engine spinning fast enough to prevent a stall. If I keep coasting and never give it gas, and reach level ground or uphill, then the car will slow down enough that the engine can't keep up, then it'll stall. But merely cutting off fuel does not cause the engine to stall.
Idle is when an engine is spinning while not in gear. That uses fuel to keep the engine spinning, and yes, if you cut off fuel then, it'll stall since the momentum of the wheels won't keep it spinning. But in a manual transmission in a fuel injected car, coasting while in gear uses no gas.