OK, many points here. I'll respond to some, some I won't.
(Graham didn't really exhibit any Dark Lord powers until after he stabbed Yoko, for instance).
OK, this one is interesting.
In the game it is never explained what his powers are, but in the light novel, Graham is said to have had the power to "perform miracles" before even getting to the eclipse. With these powers and his charisma, people started following him, believing him to be some sort of messiah. And with this, he managed to build what is described to be a religion. It grew to such
enormous numbers that these followers' bodies were used to compose an entire replica of Dracula's Castle, and there were
still a lot of people left.
So, while the specific nature of his miracles is not mentioned, it is implied he was able to do things like curing the ill.
And we don't know how Walter assumed that mantle either
We don't know even if he DID assume that mantle at all. Walter is never referred to as Dark Lord in any way.
Clearly Chaos (for whatever reason) views Soma as the ideal heir (having something to do with being the reincarnated soul of Mathias probably has something to do with it) but anyone willing to go the evil distance of evil is likely an acceptable substitute.
This bit is true. The only TRUE requirement for one to become the Dark Lord is the willingness to take over the forces of evil - namely, conquering demons.
First, we see this in Aria. Graham clearly possessed control over the castle, but he was STILL intent on killing Soma because he had "control over HIS demons". Then, in Dawn, Dmitrii pretty much confirms this in all letters, saying that the true heir to the Dark Lord position is not one who inherited powers from Dracula, but one who actually takes control over demons - with or without Dracula's powers.
So, naturally, Soma has the advantage here because he has the power of Dominance, which is a very convenient thing on controlling demons.
Now, to respond the question: I think there is no requirement beyond being born at the same exact moment of Dracula's death to become a candidate "officially". And yes, this means that there could be hundreds of other Dark Lord Candidates over the world. But, ultimatelly, what makes the Dark Lord is the control he exerts over demons.
But, of course, when you have magical powers this task becomes so much easier.
I think the powers simply attached themselves to the candidates by pure dumb luck.
What makes them special is not that they were choosen by these dark powers, but what they do with them.