Possible using the constructor:

public class Example : MonoBehaviour
{
    public Test test;

    public Test[] testArray;

    public Example()
    {
        Test a = new Test();
        a.p = 5;
        Test b = new Test();
        b.p = 100;
        testArray = new Test[] { a, b };  //your array will start with a and b in the inspector by default
    }
}

Note: using the constructor for monobehaviours is usually a bad idea, but this is one place that it'll work well, as it'll create the array before the deserializer adds in any values. It will however mean that the array is assigned twice at runtime - You can however comment/remove the code before making a final build though

No. Array values must be initialized manually. If you have Test[], the type is actually "Array of Tests", which is not the same as a "Test". Anytime you have an array, each cell has to be initialized independently. Unity handles this for you because of the inspector, so it just assigns the default/null values to the type. If you want to be able to control the values, or assign them via the constructor, you need to do something like this:

public Test[] arr;

void Start()
{
    arr = new Test[10];
    foreach(Test t in arr)
    {
        t = new Test();
    }
}

That will initialize all of the values in the array. Unity does not do this for you, as far as I know.