Analogue accelerometers output a voltage that is simply proportional to the acceleration. Add a battery, connect the accelerometer output to an oscilloscope and then shake the accelerometer, you will see the trace changing. A sustained acceleration will produce a sustained signal. The answer to yout 1st Q is 'yes.' Most would easily measure the acceleration and deceleration of a dragster.
Note that gravity is just an acceleration and so the accelerometer will show a constant acceln due to gravity (unless it's in free-fall). For this reason they can also be used to measure tilt.