With the details of the Gambit still unknown, I can easily see the other Karlsson daughters assuming that as their sister, Olivia would be added to the Board at some point as part of their Father's Last Will/Gambit.
No, just no. Olivia replacing someone is a possibility, and obviously it would change the balance of the vote. But there's just no way for Olivia to be added to the Board. It would lead to an even number of members, what is a big "no" ; at least until someone come in position to have a decisive vote.
She could have been the deciding vote that pushed the previousy 3-3 Board split in their direction.
There's already a deciding vote since the Board have an odd number of members. The tie and unknown last vote are here to serve the story ; "it's an undecided situation and the final choice will be yours".
But here, there's an assumption that shouldn't be made. It's that the two questions, "who will be the parents of Alexander's grandson" (the part of the Gambit we know about), and "who will be the next CEO of the Karlsson Group" (the vote), are necessarily linked. While talking about the Gambit, Veronica made clear that
if Kane is worthy of power, he shall have the company, and that in the case he isn't,
he will be nothing but a slave of the mother holding true power. Both make the fight for the succession totally useless.
If you take care to Veronica actions, in her mind it's clear that, if he's worthy, it's Kane that will decide of the winner, not a vote.
When talking one to one with him, she offered herself, as are doing Dominique and Juliette in the same situation. For Dominique and Juliette, the goal is obvious. Having Callista son with them is a step forward to the victory. But for Veronica it have another meaning, because she's not running for the seat of her father ; she offered herself as an option, and if Kane decide it, she'll be the winner.
It's less clear if Kane is not worthy, but I doubt that even in this case it would be decided by a vote. At least not as Dominique and Juliette imagine it. It would be an error to believe that Olivia, Kane and Seth are here only to be tested. They also are here as tests for the others. I guess that this especially apply to Seth, but it don't exclude Olivia and Kane.
It wasn't until Dominique learned that Veronica was backing Olivia that Dominique started seeing Olivia as a real threat as opposed to a theoretical one.
Because she made a strategical error by lack of knowledge. Unaware that she is about the grandson part (it's only Veronica, Yvette, Elena, the Gambit Queen and the players, who know this), she can't imagine that Veronica is pursuing two different goals ; backing Olivia for the grandson, and Dominique for the lead of the group.
Half brother? It looks pretty clear at this point that they aren't biologically related at all.
As player you know that. Veronica, Elena and the Gambit Queen, especially if you're right about her identity, know that. But anyone else in the game, including the board members, are totally convinced that Kane is Olivia's half-brother. And when I say everyone, it include Olivia. She know that Seth and her aren't related, but clearly said that she don't know if it's because she isn't Callista's daughter, or if it's Seth who isn't.
Listen to all of them, not a single one doubt about this. Even Juliette don't know. Among everyone, she's the most likely to use this information. She love to humiliate. Throwing, in the middle of her threats, "oh, by the way, Kane isn't even your real brother", would have been a perfect argument here. It would have proved the superiority of her power, since she have an information that even Veronica's favorite don't have ; and this superiority would have gave way more strength to her threat. Yet she said nothing. Why, except because she just don't know ?
It's more than probable that all the Z ranked know about Seth, but they still haven't figured what it imply, still believing the information as it was provided, "Seth isn't related to Olivia" ; what let uncertain if Olivia is related to Kane or not. Then, with this information, what is the most probable from their point of view ?
Is it that Olivia, known daughter of Alexander, and supposedly first child of the only woman Alexander truly loved and worshiped, is adopted, or is it that it's the so unhealthy child of the so caring Callista that was adopted ?
I have no idea why you even think it's possible for there to be a different male winner.
Simply because I don't project my knowledge on the characters' mind.
It's only you, as player, who know that there's only one possible male winner. But for all characters, except Veronica, the Gambit Queen, and now both Elena and Yvette, the current batch of slaves is here to decide who will be the mate of the future CEO. And from their point of view, with the knowledge they have, it can be anyone of them. Alessandra, Dominique and perhaps also Juliette, have a preference for Kane, because they know he's Callista son, and this could give them an advantage, but that's all.
Take Dominique by example. On the slave route, she wouldn't have pressed Kane to be tough from the outside, whatever how submissive he can be in the bedroom, if she knew that he'll win even if he's not worthy. It's even more visible on the K5/K6 route, where she ask him to win Veronica's support. It's not just to gain back this support for herself, it's also with the intend to make him climb the ladder and ensure that he'll be the winner.
The same apply for Juliette, once again on the slave route. As cruel as she is, if she knew how precious he is, she would have tried to train him, instead of trying to break him by demoting him to K3, with the clear intent to totally forget about him.
But it can be assumed that Alessandra have at least some suspicion regarding Kane real role. This because she's really the one that make the more effort to charm him. What could perhaps be an answer to your question regarding her goal. While Dominique and Juliette fight to the lead of the group, she could be simply making her way to the very top, the effective control of the group ; because once again, one do not necessarily imply the other.
Veronica was clearly expecting the contents of their discussion to be leaked. So either Yvette is known to be a bit of a gossip or Veronica was aware of the room being bugged, probably by Dominique, since she's the one who reacted afterward.
Veronica made clear that among what she said there were information that can be leaked, and information that can't.
Anyway, as Yvette said, if Juliette knew about this, she would already be raping Kane. And there's no doubt that it's what she would do. Yet, on the slave route she had him in her room after the meeting, and didn't tried. The chastity cage wouldn't have been a problem, she's a board member, Delilah would provide the key if asked to ; what wouldn't be difficult, since Delilah was here.
I never said that it was certain that Callista is the 7th Board member, I simply listed it as a possibility.
I quote
you: "[...] and the mysterious Gambit Queen who is very probably the 7th as yet undisclosed Karlsson board member."
"Very probably" is more than a possibility.
If Callista isn't the unnamed 7th Board member, than internal security inside of the Karlsson Group has been heavily compromised.
Callista was a founder member. She's the only person, with Veronica, that Alexander ever trusted. She had at least one child with the only male who ever reached a Z rank. And she was able to hide from the Karlsson group for a very long time ; hiding so well that everyone except Veronica seem to believe in her death. She surely still have her way inside the Group and probably some loyal persons among the elders.
Replacing a deaf cleaner by one that hear perfectly isn't something really difficult. Not only they are totally masked, and so pretty anonymous, but anyway I'm sure that Veronica don't even look at them.
There's also the meetings. The mask are obviously here to keep the mystery for the players, since themselves know perfectly who is the other. But the recluses places for the meetings between two members of the Board, really ?
They all live near one to the others and have many occasions, and reasons, to meet every day. Plus, the masked daughter achieved to reach Kane bedroom, enter it, and leave something for him, without being noticed. It wouldn't be difficult for her to meet the Gambit Queen in her home if she was a member of the Board.
Therefore, why meeting in recluses places, where it would be suspicious if they are caught, when they have so much possibilities to meet without raising any suspicion ?
Yes, but the private conversation between her and a dom Kane in 0.5 was kind of sad. She appeared to be having a bit of an existential crisis. It's certainly possible she was faking it, but Dominique suggesting she was thinking about quitting her struggle to improve the world which has gone nowhere to focus on her own pleasure instead was a little depressing.
And what if she was both sincere and faking it ?
Sincere because yes, she pass much of her time caring about the other, and with such sisters she can wonder if she isn't missing something they have in their selfish life. But also faking, because she wanted to test Kane, and see if he can be a good fit for her ?
As I said, so far what she know about the situation is that the male and female winner will end together. She support him for who his mother is, but what she expect from her future husband goes further than pure entertainment ; she have an endless supply of slaves for that.
Alessandra is arguably the biggest sadist of the bunch, but she's not impulsive, nor does she engage in bullying power plays like Juliette. [...] She's sweet to those she likes, or at least appears to like. Heck, she even talks sweetly to those she's torturing but that sweet talk doesn't stop her from torturing them.
She's not impulsive specifically because she have a higher morality sense than Juliette. She's sadist, but not cruel, that why she can have a sweet even for someone she's torturing. Unless she's doing it because she think they deserve the wrath of her justice, she genuinely assume that they are enjoying it as much as her.
She probably only need a (relatively) small push to be put on the light side, and only express her sadism on person who are effectively masochist.
I appreciate the hardwork you do and am in no way trying to hurry you 
