I highly doubt that the Fallen and Hive are related in any way. The Fallen do not directly serve the Darkness, for they fight the Hive--and the Hive embrace the Darkness and revere it.
The Hive are undead. They are not an infection or biological entity like the Flood from Halo. They are necromancers who have used the power of the Darkness to keep themselves alive for eons--they are directly inspired by fantasy undead. The location of "The Summoning Pits", and the mention in a patrol mission that "space barnacles" are biomass grown by the Hive makes me think they grow corporeal forms from this biomass, which they in turn bind summoned souls of old to. They are an ancient monarchy that has preserved their hierarchy through millennia of undeath. Thralls, Acolytes, Knights, Wizards, Ogres---and perhaps more enemy types we haven't seen. They are a race of liches.
Why are they undead? Perhaps they hit a snag in their own evolution when they were truly living, and turned to the Darkness to preserve them as they saw it as the only way to escape extinction. Or, perhaps they were so corrupt and greedy for eternal power when they encountered the Darkness that they simply embraced it.
The Fallen on the other hand are completely biological. They don't show any signs of being undead despite their souls escaping from their bodies when killed. They had a grand past which they scarcely cling to; they are now nomadic barbarians that pick apart whatever they find. They assert what's left of their former authority and nobility onto whoever gets in their way, for they have claimed all of our former worlds for themselves. They think they are still entitled to riches and glory, and since they have none left, they resort to forcefully taking it from others. They are divided amongst themselves into five Great Houses.
Unlike the Hive, their physiology is different. Their limbs may be similar, but their head shape and eyes are very different from the Hive. The Hive come in all shapes and sizes--the twisted frames of the Thralls, the hunched-over Acolytes, the upright (and tough) Knights, the floating Wizards, and the Giant Ogres. But the Fallen (apart from their servitors) have the same basic shape across their ranks. Also, it's highly likely the Fallen's current nomadic raider behavior is a much more recent development than the Hive's necromancy.
I still don't think there are enough similarities, even for them to share a common ancestor. Their origin stories are just too different.