OK hypothesis time: Galamoth is a demon originally from the future, who sent the Time Reaper to 1797 to destroy Dracula during the events of SotN. The Reaper was intercepted by Aeon and failed, so Galamoth gathered his troops and went back in time himself, but also failed by being destroyed by Alucard as soon as he got there.
Now for addressing the points:
1. The Time Reaper.
Judgment is not given a proper time period. As the game states, it happens inside a time-rift created by Aeon that helps bringing people from everywhere together. Yet, the Time Reaper says that he was sent from "10,000 years hence" meaning that the Time Reaper clearly has an specific time period where he must land from his travel. If he can count "how many years he has been sent from", then, at least the place he is in, HAS a time period. One more evidence for that is that Aeon says that where the Time Reaper is, only one person can go -- implying that this space is a completelly different space from the Time-Rift.
In the end, the Time Reaper is destroyed, and calls for Galamoth's name as he disappears. While the game didn't tell us where in time it takes place, keep the Reaper's "10,000 years hence" bit in mind.
2. Galamoth went back in time himself with his troops after the Time Reaper's failure.
Galamoth's japanese description says that he is "a demon with a 10,000 years plan to take over the Demon Realm." This, coupled with Judgment make it obvious that Galamoth is nobody's servant and wants to destroy Dracula.
Beyond Galamoth, Frozen-Half is also not a servant of Dracula, being a servant of Galamoth. Another enemy present in the Floating Catacombs is Salome, whose japanese name is "Sausine" and japanese description says that her nickname is "Ma-chan". "Ma-chan" is an enemy from Boku Dracula Kun who is also a servant of Galamoth. After SotN, the same enemy sprite returns, but is called simply "Witch", with its description saying she's nothing more than a common witch.
So, while Frozen-Half is confirmed to serve Galamoth, Salome could possibly be his servant too based on the details I mentioned. Frozen-Half is never seen again in the series and neither is Sausine (as Sausine), implying these could be unique, irreplaceable enemies.
We know that Dracula is the Demon King and has Dominance over demons. So why aren't Galamoth and Frozen-Half (and arguably Sausine) under Dracula's command like every other demon in the castle? Galamoth is an extremelly powerful demon, so Dracula would never overlook that!
The explanation is really simple: Dracula simply does not know who they are. And this is possible if they JUST started existing in SotN -- by having come from the future to 1797. And I can prove that too: In the last battle against Aeon in Judgment, if you're playing as Dracula, Aeon says quite clearly that Galamoth is hunting Dracula from 10,000 years in the future, and Dracula's response is: "I fear no one requiring such absurd means to face me." Dracula has NO CLUE who Galamoth is. This is only possible if Galamoth was born AFTER Dracula's time.
As Aria has shown, the situation changed, though, and Galamoth's soul is under the castle's control by 2035, trapped in a candle. This could be a result of Alucard destroying Galamoth, and his soul becoming susceptible to the Castle's influence, as Eric Lecarde explanation about stray souls goes in PoR.
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Now, remember the "10,000 years" thing? So, Galamoth has a "10,000 years plan" and the Time Reaper comes from "10,000 years hence"? Coincidence? I think not. I think sending the Time Reaper was the original "10,000 years plan" and, after it failed, Galamoth travelled himself 10,000 years in the past to try to destroy Dracula.
Here are some interesting details:
-Galamoth comes from Kid Dracula, and most fans know this, but get this: Kid Dracula takes place in 11,797 -- exactly 10,000 years after 1,797, the same year of SotN's events.
-Galamoth's soul in Aria has the power to stay unnaffected by changes in time, explaining maybe how he was able to travel back in time.
-In Boku Dracula Kun, Kid Dracula is assisted by Death, while Galamoth has his own parody version of Death -- Shinigami-san. What does Death say to the Time Reaper when he meets him? "Even your visage mocks me. Die!"
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So that's it. While the hypothesis explains where does Galamoth come from, it doesn't explain why his objective was 1797, however. BUT I have one last hunch: I think the issue with time that Saint Germain was sent to fix on Curse of Darkness has to do with Galamoth successfully erasing the chunk of time where Dracula existed.