You're correct that each fragment will have its own independent orbit, based on the delta-v that it gained from the explosion. But they won't circularize; each of those new orbits will intersect the original orbit at the point where it exploded. The orbital periods of the fragments will also not match the original, meaning that the relative intersection will shift along the original orbit every time. So an explosion in geostationary orbit would be a Really Big Deal and could potentially threaten the whole geostationary fleet.

Orbit can be both higher and lower, actually, any deltaV except purely in the axis of movement will result in exactly that.

Yes. you can accelerate away from the Earth and have a lower periapsis (conversely, you can accelerate towards the Earth and have a higher apoapsis). The only way you can maintain the same periapsis is by accelerating in the same direction you're going. Highly impossible in an explosion.

Also note, it is impossible to raise periapsis in a single event (or single time frame), except by leaving Earth orbit entirely (i.e. eject to planetary space)