This is quite a hard question, so I will do my best to summarize what I've found on Orcus and Demogorgon. Also, the 4E Book of Vile Darkness has a lot of information about demons lords.
Orcus
- Lieutenants: Vampires, liches, necromancers, cult priests
- Minions: Undead (ghouls, shadows), cultists
- Races: Not relevant for undeads, humans, orcs, half-orcs, ogres and giants
This may be the easiest one, because of all the detailed description about him. From the MM p. 52:
Orcus is worshiped by the undead and by living creatures that channel the power of undeath.
Which makes the choice for a lieutenant relatively easy. On multiple wikis (like here or here), liches and vampires are mentioned as his most powerful worshipers. In the 4E published adventure Keep on the Shadowfell, a human priest of the Cult of Orcus tries to open a portal to the Shadowfell. Also, from the MM p. 53:
Orcus, the Prince of Undeath, has the power to transform manes into undead monsters, most often ghouls and shadows.
We can assume he does that to make them serve him as minions. From the Book of Vile Darkness p. 138:
In particular, orcs, half-orcs, ogres and giants revere Orcus, as do a large number of corrupt and despicable humans.
Another quote from the BoVD p. 138:
Intelligent undead never willingly serve Orcus (they are more likely to venerate the deities Vecna or Erythnul). However, many vampires, liches, and other undead creatures are forced into his service by dark pacts or compelling magic.
Demogorgon
- Lieutenants: Severik the balor (mentioned in the BoVD p. 129)
- Minions: hezrous, aboleths, scrags, skum, various ichytoid beasts (BoVD p. 125)
- Races: tiefling, lizardfolk, half-elf (examples from the BoVD), kuo-toa (Out of the Abyss), Derro (Out of the Abyss). Mostly any kind of creature that has reached a pretty high level of madness.
Not many quotes for the Prince of Demons, but you will find more details in the BoVD.
Other demon lords
Well, when I started writing this answer, I thought I could gather information from websites, books and the like, and summarize my findings here. It turns out, the Book of Vile Darkness from 4E is really the best source of information about the demon lords, so you should probably buy it for the other ones.
RAW: the barbarian will be disintegrated
The rules now say this explictly per the 2018 PHB errata to close what was previously a loophole.
The target is disintegrated if this damage leaves it with 0 hit points.
Since, the barbarian would be at 0 HP after taking the disintegrate damage they would be disintegrated.
The loophole was never intended
Jeremy Crawford, lead rules designer for 5e, confirms this loophole while also confirming that it was unintentional in this tweet (made before the 2018 errata):
The disintegrate spell has a loophole in it. RAI: The spell disintegrates you whether it deals damage to you when you're at 0 hit points or it reduces you to 0 hit points. RAW: You aren't disintegrated if the spell deals damage to you when you're already at 0 hit points.
Best Answer
It's up to the DM.
Usually, we would apply the specific-beats-general rule to work out contradictions:
The trouble here is that both Demons and the disintegrate spell present specific contradictions to what generally happens when a creature dies. They are both specific rules. Here is the general rule for NPC creature death:
This is the general rule. The creature dies, and there are no restrictions on resurrection, the body is just a body, etc. The usual, expected stuff when you stab something to death.
Demons create an exception. Instead of just dying and leaving a corpse, we see, in the section Eternal Evil:
Disintegrate also creates an exception to the general rule, but creates a contradiction when used on demons:
Because these are both specific rules that contradict the general rules for dying, one cannot clearly be seen to be more "specific" and the other more "general". These are both specific rules that contradict one another. It is up to the DM to decide how this works.