Why do the supposed dragons look pretty much exactly what were called wyverns in the previous installment? Was it an art change, or do the people of the time of drangleic simply call wyverns dragons?
It's probably just the people calling them dragons. You have to remember that the world went from a place with absolutely no dragons to an entire shrine and aerie dedicated to them.
Why does vendrick's soul have the same icon as a lord soul? Should he not have a dark soul, being a man? He did absorb the four lord souls as the cursed undead, but since those four are up and about again at the time of the story, one would assume he no longer holds any such lord souls.
First of all, I'm not sure there are any "Lord Souls" in this game. There are Great souls, definitely, but no "Lord" souls. From my understanding, the four who had these souls before you wrenched them from their lifeless bodies were extremely old and extremely powerful, but not actually unique like the Lord Souls were. Think of them as an undead (or otherwise) that had grown extremely powerful and extremely large over time, but not by the graces of a deity, but solely through tenacity. I'm almost certain that's what those souls entail. Vendrick's soul was large itself and powerful itself because he had amassed a large amount of souls and had become extremely powerful in his own right.
On the note of Vendrick, I'm pretty sure that when you fight him in the Crypt that he's divested himself of his soul and then retired to the crypts voluntarily. I think that the Gravewarden merchant in there actually says something to that effect anyway (Namely that he was there voluntarily). Also, if I remember correctly, everything talking about Vendrick that I can think about denotes him as a good man and a great king, not in any way a part of the dark. Nashandra has been repeatedly referenced to manipulating the King, and using him for her gain. Which if you read some descriptions, she desires the Throne of Want more than anything else. That leads off to the Throne of Want and all it's fun notes, so I'll plop my understanding of it here.
The Throne of Want, by my estimation/guesswork (I'm probably off), is another incarnation of the First Flame. It has been guarded by the Giants since as long as anyone can remember. Nashandra found out about the Throne and manipulated Vendrick into fighting the Giants to steal the throne for her. Vendrick crossed the sea to a different land, incurred the wrath of the Giants, and returned with the Giant's most prized possession, the Throne of Want, an incarnation of the First Flame (I'm going to have to remember exactly where I saw the reference to the flame, I'll go search again later). From the final sequence in the game, it tells you that the Throne of Want shows the one who sits upon it anything they most desire, which sort of leads me to suspect that the fire changed over time as a way to prolong it's life span, slowly siphoning away the soul of the one who sits on it, but never allowing them to leave, instead of incinerating instantly the person who kindles it.
And since I was talking so much of the giants a second ago, time to step into speculation about the Last Giant and the Ancient Dragon. You mentioned that you thought that the Giant Lord was the one who was trapped underneath Cardinal Tower, and I'm not entirely sure that's correct. I think there were many excessively large giants, like the Giant Lord was, and that they were all iconic figures of the Giant society, but I don't think the trapped one is the Giant Lord himself. I'd actually sort of speculate that the bodies whose memories you enter are the giants themselves, and that the one where you fight the Giant Lord is the corpse of the Giant Lord. That doesn't make too much sense with looting the corpses of giants and gaining their souls, and then getting punted out of the memories, but those giants aren't the same size as the
withered corpses of the giants you enter. (Also, off of the top of my head, the reason why you don't get a giant soul from the Giant under the Cardinal Tower is because he's been trapped near death down there for hundreds of years, his soul's probably been degrading over time. All of the Giant souls you get from the memories are from Giants that are only shortly dead, excluding the Giant Lord, but he doesn't technically give you a giant soul).
The Ancient Dragon uses magic to let you enter the memories of centuries dead giants, something which he probably wouldn'tve been able to concoct by himself if he were just surrounded by dragons and the drakekeepers or the covenant members. I'm gonna go out on a limb and say that he probably lived on the other continent with the giants at some point, then he migrated to Drangleic at some point. If you talk to Nashandra after you get the Ashen Memory(? I can't remember the exact key description), she says to not listen to the dragon, that he is a "false god." I'm thinking that the large dragon that died in the Brightstone Tseldora was the original "Deity," and that the Ancient Dragon sort of slid in to compensate for a lost god. I think this also makes a sort of sense if you think about the large dead dragon in the Tseldora. When you enter his memory, he's almost immediately dead, his soul up for grabs. The area is open and largely reminiscent, to me, of the Dragon Aerie.
I'm going to take a break for now and probably paste some more answers after criticism/running around some more.