This was done using pure physics as one contiguous sequence, no baking, no faking. If you want to make things a little easier, baking will allow you far more leeway when dealing with finnicky physics.
The Blender version is 2.77a, but it should behave similarly in 2.8+.
(Blend file below)
When put together the ball would not behave as required, nor would it sit properly in the dented cube when a bounce setting was added. As expected it would not fall to the ground after the explosion either.
The visible cube and ball in layer 1 were therefore purged of physics and a duplicate cube (in layer 2) added and given rigid body physics (passive) instead.
To bend Blender's physics behavior to what is required A 'Physics Box' was also used. (PH-box in the Outliner)
A Ph-box is simply an invisible substitute rigid body that can be any shape or size and located anywhere with respect to the ball, so the physics engine acts on the box, not the ball. The ball is parented to it outright. (CTL-P)
The box's Origin Point can also be located anywhere, giving you quite remarkable physics flexibility when compared to limiting physics to the ball itself.
It might be possible to obtain the behavior required without the PH-box but it's a lot easier with it.
The PH-Box is located at the lower surface of the ball.
The box falls to the cube as expected.
As is so often the case, when bounce and other factors are introduced, things can go awry as it did here so the ph-box & duplicate cube's bounce settings were keyframed to suit each stage of the sequence.
The duplicate cube was also lowered sufficiently to have the ball settle into the visible cube's indentation without needing to fiddle around with margin settings too much.
When the explosion occurs, the cube in layer 2 is simply keyframed (instantly) below ground so it's out of sight. The PH-box has no choice but to fall, taking the ball with it.
The box's bouncing was enhanced a little at ground level so it can be seen amid the debris flying around. (keyframed in the PH-box's physics settings)
Also note, with a PH-box, the ball (the child) can be animated slightly to shift lower on the way down so it's surface is at true ground level, negating the need to fiddle with margin settings etc.
Another version of the same thing, but baking physics at each stage, giving far more control over the ball's behavior...