So, why exactly is it that the serpent is capable of moving across the celestial sphere? What makes the lights of this constellation different from the millions of other holes in oblivion?
The Serpent is made of unstars. That is, it isn't made of stars. So it doesn't need to behave like other constellations.
And while we're at it, can anybody explain the sun and it's funky heliocentric path when it should by rights be painted on the celestial sphere?
I'll explain: the sun
doesn't have a heliocentric "path" because "heliocentric" means relative to the sun. In other words, the sun doesn't move relative to itself, nothing funky there.
What I suppose you meant to ask is why the sun moves in a nirn-centric path (relative to Nirn, duh) when it's pinned to the celestial sphere. Being pinned to the celestial sphere only makes it "funky" for objects in the sky to move relative to each other, like the Serpent does relative to the stars, questioning its state of pinness. There's nothing in lore (nor in common sense) that demands that Nirn should also be pinned to the sky so that the sky, sun and stars shouldn't move relative to it.