There are no huge errors, although I have to say I am not a fan of the Lex parts. I guess he is without question a villain. It makes it harder to see why Lois ever fell for him in any life, but I guess he is also very good at faking a situation.
Lex patted his double on the shoulder.
Too bad that Clark does not realize Lois saw clones in her memories. I guess even if he did it would not help much here.
“You’ll appreciate another visit topside, this time with some fresh air,” he said, leading the clone to the secret elevator hidden behind a bookcase in his ark’s study, where Nigel was waiting.
I guess it makes sense to have the clone in the Ark.
Lex inserted his keycard and typed in his code, the date of his White Orchid Ball when he had first met Lois.
He is obsessed with Lois more than I suspected.
“Nigel has an evening of skeet shooting planned for you, should you do well with your lessons.”
That was an interesting scene, with Lex skeet shooting. It was one of the signs that Lex really did not care about others than Lois ignored.
The Lex-Clone cowered for a moment before his resolve stiffened. “Well, it wasn’t as if I was speaking to an underling, Father. Should I not address you with more respect?”
I have to say this is better than the interaction between the clone and Lex in Vatman, even if the story of Little Red Ridinghood insurance fraud was funny.
She’ll be teaching you about proper tables manners and decorum while in the company of women.”
I think the phrase is usually "table manners".
The clone’s brow furrowed. “But, Father, if might is right, and the Luthors are mighty, why do I need ‘polite manners’?”
Considering how the SM clone treated Lois, I think in canon Lex never addressed this issue at all.
Certainly not fully ready for exposure to outside world as Lex’s double.
I think that should be "to the outside world".
“Yes, but do you really want Mrs. Cox in our little secret?”
I think that should be "in on our".
“Don’t call her that!” Lex hissed. “She is Dr. Carlin; that is all anyone ever needs to know. I had to inform Ari because her viewing room of the psychiatrist’s office is on the other side of Lex-C’s bedroom.”
The current phrasing probably works, but I half think it more likely he would say "the viewing room of the psychiatrist's office" or use her to modify both room and office.
If Lex-C had no personal experience with women, it was conceivable that one would be able to seduce him and thereby turning him into a docile and blathering idiot,
I think it would flow better if it was "thereby turn him into".
No, Mrs. Cox would be perfect tool for Lex-C to learn that women were objects to be used for nothing else save enslaving,
I think the phrasing should either be "a perfect tool" or "the perfect tool". Luthor has a truly disturbing view of women and how to interact with them. I guess we have seen this to some extent before, but not how deeply disturbed he actually is.
Starting Lex-C out with a female toy, whom he could enjoy by also physically hurting, would show the clone the true pleasure of women.
No wonder Clark does not trust Luthor.
“I won’t be staying the night, Asabi, and neither shall the lady,” Lex corrected.
Spending the night and using the bed and two different things, and the one does not require the other.
“Have precautions been made to ensure that the others in the ark don’t learn of my attentions? Each lady needs to feel as if she, and she alone, has my undivided attention.
The only way to maybe achieve this would be to isolate the women from each other. As long as they speak with each other it is highly likely they will mention his attentions to the other women.
“Please, call me Brenda,” she insisted. “Shall we now retire to your bedroom? Or do you wish to get undressed in here?”

not beating around the bush here.
“Where would you like me to examine you?
Interesting development.
feeling distinctly outside of his comfort zone and taking a step backwards away from her.
I think that was her goal, not only with this line but with showing up early.
She wasn’t acting and speaking as he was led to believe a woman of her occupation and station would.
This is Clark's major advantage over Luthor. Luthor thinks the position defines the person, and as long as he sees Clark as merely a reporter, he will not at all suspect that CK=SM.
“Please, Lex, if we’re going to be intimately acquainted, you must call me ‘Brenda’.”
Hmm, she does not believe in beating around the bush.
my guess is that you want to fill my uterus for the coming dawn.”

She really does not beat around the bush.
while I’m not a fertility specialist, per se, I do have lot of experience in the field.
That should either be "have lots of experience" or "have a lot of experience". I am not sure which form is better.
If any of the women in our community seem difficult to impregnate within this first year,
I find Muldoon's clinical approach to these issues disturbing, even if I guess if they were really all that was left of humanity it would make sense.
For the continuation and health of our species, it’s a must.”
I really hope the prison gets broken up before then.
beginning to wonder if he had misjudged this woman completely.
I am surprised there is still any doubt. Although considering what she things reality is, her actions are mainly logical. Although I would only admit that fully if the reality of the ark is significantly more females than males. Even if it was, I still would not wish being Luthor's child on anyone, so I still would dislike Muldoon's plan.
“Third session?” Lex gulped. Was she serious? “Tonight?”
Well, Muldoon is doing good at keeping Luthor on his toes.
He bristled at the assumption that he would need any pointers,
I am glad someone has managed to insult Luthor in a new and unexpected way.
He finished off his Scotch and set the glass on the sideboard as she continued.
He is not doing well at listening to the doctors instructions.
if only the asteroid had struck Earth, and Lois had willingly agreed to join his ark community.
Lex needs to realize the failure of Lois to join him is not conditioned on the asteroid not striking.
Not only would he have a harem of women ready to bear his children ‘for the good of mankind’, but Lois could now be his queen to do with as he pleased.
Lex really does not understand Lois if he thinks she would ever agree to be with a man who had a harem.
Lois would be joining his harem, by force if necessary.
More evidence that Clark is right to not trust Luthor.
It was that flying menace’s fault that his ark was only a fantasy of Lex’s concoction, instead of the real deal.
Except with no SM, the asteroid would hnot have hit at all.