Distance between Object 1 and (Any part of Object 2, not center only)

Questions & Answers
1

How can i calculate the distance between object 1 and the entire area of object 2?

2

You could do it with a selective raycast and then getting the distance of the hit point from object 1 (object 2 needs to have a collider on it).

int object2OriginalLayer = object2.layer; // store original layer of object2
object2.layer = specialLayer; // put object2 on a special layer so only it will be hit by the raycast
Vector3 object2Direction = object2.transform.position - object1.transform.position;
RaycastHit hit;
if (Physics.Raycast(object1.transform.position, object2Direction, out hit, object2Direction.magnitude + 1, 1 << object2.layer)) {
    float object2PartDistance = (hit.point - object1.transform.position).magnitude; // distance between object1 and the point hit on object2
}
object2.layer = object2OriginalLayer; // restore object2 to its original layer

Note that this will not always hit the closest point on object2 (if my geometry is correct), but if object2 has a convex collider, then it will be close.