Understanding an N-JFET
Why is Vgs important and how do we calculate it?

As written in the answers and comments: Vgs is not something you calculate. Vgs is an external property (a voltage) you apply. Vgs is important because it determines the behaviour of the device, i.e. how easily it will conduct current.

Compare Vgs to the gas pedal in a car. It determines the speed of a car. But it's not a one-to-one relationship. A car with an empty gas tank will not move. A car with a strong headwind or up a hill will move slower.

How can we calculate the pinch-off voltage without the datasheet?

We cannot. You could measure it though. Apply a certain Vds and a certain Vgs. Then vary Vgs and write down the value when the current from the Vds source becomes zero.

In the car analogy it is like asking how far you need to push the gas pedal in to drive up a hill at 10 meter/second without you knowing anything about the car.