For your first part, if an object is floating then the mass ( or more precisely the weight) of the displaced fluid must be equal to the weight of the object. This is archimedes most famous contribution to science.
If this is not true the object would either accelerate upwards or downwards. It would do that until the condition is true or the object sinks.
In the next part where you give 1000 * 0.2 * 0.2 * h you are calculating the volume of the cube under the water. i.e the part of the cube that displaces the water and lifts the cube upwards.
So the h is the amount of the height of the cube which lies below the water.
Which of course leaves the remainder ( 0.2-0.13 = 0.07) of the cube above the water.
Note that you give 0.7 as this value but it is really 0.07.
Once you add the lead you are required to do this UNTIL the block is completely submerged.
So the volume of displaced water is then 0.2 * 0.2 * 0.2 Just as you have done.
And the mass of the displaced water is 1000 times this. ie the mass of the water is 8 Kg
It is capable of supporting a total weight of 8 Kg. But the block only weighs 5.2 so if you DIDN'T add any lead and you let the block go it would pop upwards like a cork.
But by adding the extra 2.8 Kg of lead the total downwards weight is 8Kg and the total upthrust caused by displacing the water is also 8Kg. As they are in balance the block will stay where it is put. Just at the surface of the water.