We find ourselves on a planet very much like earth. Suddenly out of nowhere disaster strikes and an annihilating explosion on planetary magnitude leaves the poor thing with half its mass practically evaporated away, leaving no debris and for some reason very little damage on the remaining part.
Storms, earthquakes and disappearing oceans
Even on the furtheres points away from the disaster, powerful earthquakes will shake the ground as the planet is adjusting to its new gravitational center. Storms will be building steadily as the atmosphere is rushing to fill the gap. People living by the ocean will soon begin to see the waterlevel drop slowly but surely. If they have the time to notice such things amidst the building storms and quakes.
All things bad
Depending on which part of the planet was exploded away the path it travels through space will shift. I'm not sure in which way, so let's leave that for now.
If the remaining half was left unhurt by the explosion itself it will not be so safe from the planetary scale event that is going out of control inside it. What once was the gravitational core of the earth is shifting towards what makes up the center of the remaning half.
This is where the remaining solid inner core wants to travel. (or you could say that the remaining planet is being pulled around the core, but the effect is the same) And in doing that traveling it will be displacing what is left of the liquid inner core and burrow into the mantle.
The displaced liquid core will now spill out over the edges of the
mantle and crust in tandem with the edges of the crust and mantle beginning to collapse under the gravitational forces.
The lower and more liquid parts of the mantle will move inwards towards the new gravitational centre and as this happens the outer mantle will begin to break apart into fractions. When this happens there is no hope of saving the crust and what might live on it. Volcanic eruptions will fire in the faultlines following the fractures. While this is happening a lot of the atmosphere will burn and evaporate away.
When the planet stabilizes around its new gravitational center it has been reduced to an uneven ball of fire and smoke. Any life we would recognize from Earth would now be dead and most organic remains burned away. Who knows when life might thrive again, if ever. Depending on the new path around the Sun, maybe never.
Edit in response to edit of question
the initial collapse of the remaining planet doesn’t necessarily form a regular core (after all the earth itself is somewhat irregular) but whether it reshapes into an approximate sphere in time will to depend on how much was left of the planet after the explosion. The bigger it is the more likely it is that gravitational forces will work it into a sphere, but the smaller it gets there is a point where the gravitational forces of the core would no longer be sufficient to alter its outer layers. I’m not sure what the limit is here, but I would guess it would be rather small. Ex, look at the moons orbiting mars, or look at the dwarf planet Ceres.
As an irregular shape it would most likely keep more regular on the areas furtherest from the initial event and more broken and flat on the side of the event. It might look somewhat like a sea urchin at that.
As for whether it would break down into chunks I would assume that to be unlikely if the event initially left the remaining half of the planet relatively unharmed. If this was not the case and the initial explosion did impact the remaining half with considerable force then it is more likely that the remains would break into bits. In which case a lot of these bits would be liquid or molten rock bobbing and bubbling through space as they slowly cool down.
However, unless fractured and broken by the explosion itself I would assume the gravitational forces of the remaining body would be too great for it to simply crumble into pieces that drifted away from each other.