Hello
1. the transistor is to be in the saturation region when VGS ≧ Vth and VDS ≧VGS-Vth. Look at the condition AND
the saturation region itself the transistor can be under the strong or the weak inversion (also called subthreshold) depending on the difference value of VGS from Vth. it is assumed ideally that if VGS is less than Vth the transistor will not conduct a current, however, in reality the transistor starts to conduct small amount of current before this Vth, the current is referred as the subthreshold current at the subthreshold voltage . I attached you this image, it will help you.
2. I think you are considering the transistor switch charging a capacitor,;-). yes , it is reffered for the saturation region.
3. exactly right, see also my image