Concerning Dior and the House of Bëor:
The House of Bëor goes into a diaspora situation at the Dagor Bragollach. They flee Ladros, and the refugees travel to Brethil (and later Dor-lómin). In time, these refugees would go from being 'foreign guests' to being members of those people. Those living in the Forest of Brethil would be under the rule of Haleth's heir, and those living in Dor-lómin would follow Hador's heir (at this time, Galdor).
Barahir is thus the last chieftain or ruler or lord of the House of Bëor. He was Lord of Ladros...but thanks to Glaurung, Ladros is a desolate ruin. Beren, his son and heir apparent, never claims that title or role. He abandons Dorthonion and lives as a loner (or with Lúthien) for the remainder of his days. He never 'reconnects' with his people. The only instance in which he claims that inheritance is...when he goes to Finrod with his father's ring. That is his claim to authority, not among his own people, but to the elves. Finrod acknowledges Beren as Barahir's Heir when he agrees to undertake the quest with him.
Dior is born in Ossiriand at the end of Season 6, after his parents return from the dead. There is no indication that Dior has ever travelled to Brethil or Hithlum. Nor even that he has met any other mortal humans besides his parents (though we could find ways to introduce him to them if we wanted to). So, no, it would not be a 'natural career path' for him to become leader of a scattered people from a conquered and haunted realm that he has never met. His father walked away from that path, and so there is not a 'path' for him to follow. The only connection he has to that heritage is the Ring of Barahir.
So, if we want to show Dior honoring his father and his mortal heritage from his paternal side, we should depict him wearing the Ring of Barahir. That's it. That is the beginning and the end of his responsibility in that regard. [As for his nature - Dior is indeed mortal, marrying when he reaches human maturity, not elven maturity. So it is no mystery to him or his people that he has been granted a human lifespan - his swift childhood revealed that.]
As for 'Eluchil' - obviously, this is not a surname, because Elves don't use surnames. The sobriquet means 'Heir of Thingol,' and is a reminder that he has the right to the throne of Doriath. Thingol died, leaving behind no son. His daughter Lúthien has shown no interest in ruling, and no longer lives in Doriath. So, inheritance passing through the distaff line is not that unusual, though we have established that Elves have rather unique views on inheritance, not necessarily expecting to need an heir. Dior, on the other hand does expect this - he names his sons Eluréd (Heir of Thingol) and Elurín (Remembrance of Thingol), making their roles as suitable future kings clear in their names. And, naturally, it is in honor of Thingol, which...brings us back to this season.
Concerning Thingol in Season 6:
Naturally, Thingol does present an obstacle to the marriage of Beren and Lúthien. He disapproves, and does not want his daughter to marry a mortal. Rather than flat out forbid it, he proposes the quest, basically telling Beren to 'go away'. This is already a mitigated response - he had wanted to imprison or kill Beren for entering Doriath unbidden. So, in the proclaiming of the quest, we can depict him giving a more measured response than his first impulse, just as we saw him give a more measured response to the kinslaying reveal than his first impulse. He issued the Ban, but did not declare war on the Noldor or keep Galadriel and her brothers imprisoned.
So, what should happen, is that the episode in Season 6 should have some parallels to the Episode in Season 4. Thingol is angry, he talks the dilemma over with Melian, and then he issues his decree. But...there are some changes. It's not simply a repeat of the earlier situation. His willingness to listen to Melian's counsel is reduced, and his anger over the situation should seem a bit more disproportionate. In Season 4, he has just found out that his brother and niece were murdered. His anger against the Fëanoreans over this seems very understandable. In Season 6, the borders of his land have been violated, so he is justifiably upset over this (he did ban humans from entering Doriath, after all, foreseeing it would bring about his people's doom). But now, the difference is that his daughter loves this person, and he's still treating him as an enemy. So, his anger in this situation comes across as must less justifiable.
That's okay - Thingol has hubris, and it is going to bring about his downfall. Failing to listen to his very wise wife is the root of his problems, and we can show that without destroying his character. Quick, unreasonable actions aren't great leadership. We don't need to frantically polish up his character to preserve his reputation - people can see the flaws for what they are.
What we want to do is to contextualize his decisions so that they don't come across as needlessly cruel or completely unreasonable. He is going to imprison his own daughter. That...is not a good look. But if we depict his very real fear for her life and safety and well-being as the motivation for that decision, he comes across as an overprotective father, not a deliberately cruel and unfeeling father. Same with his decision to put a stipulation on his daughter's marriage - the quest can be portrayed as his answer to the dilemma he is placed in. It gets Beren out of Doriath as quickly as possible, without directly causing any harm to him. Thingol supposes his hands are clean, and his land and family are safe. He is trying...though as Melian will be quick to point out, he's not as clever as he thinks, and trying to control this situation will not end how he wants it to. It is okay if the audience judges him to be a jerk. It would not be okay if the audience views him as completely racist and unreasonable as well as being a jerk.
The best chance of reconciliation comes when Beren and Lúthien return. Beren declares the Quest fulfilled, but, well, he didn't exactly bring the silmaril back with him. Thingol chooses to accept this as his conditions being met, and welcomes Beren as a son-in-law who may now live in Doriath. That is...a significant change from their first meeting. So...something has happened in the interim. Thingol has gone through some things that have taught him that his first response was too rash. What? He lost Lúthien. Lúthien left Doriath, left the safety of the land where she has lived since the end of Season 3, and marched directly into danger - facing Sauron, Carcaroth, and Morgoth (okay, and not to mention the sons of Fëanor, who also have an interest in the brideprice Thingol named....) Thingol's time bereft of his daughter and fearing for her safety has obviously mellowed him...which means he has taken the time to reconsider, and recognize his own role in what is happening. Perhaps a 'you were right/I told you so' conversation with Melian, where he just wants his child back, and for everyone to live happily will show a bit of humility on his part, and make him understandable as a father, not just a king.
We will have future opportunities to mitigate the harshness of his character with the fostering of Túrin...but he is going to end his life alone, being slain by dwarves he was treating maybe-less-than-fairly (depending on which version of that story we follow). At the end of the day, he is a jerk, but it's okay if his death is viewed as a tragic end of his life, with might-have-been possibilities glimpsed by the audience of how he could have gone differently.
