As I said earlier, Rufus can drop (old fashioned) insults but he isn't gonna vocally drop f-words or curses in gen unless he spend a lot of time with the likes of Elena,Reno & Rude & start picking up some of their speech patterns. Rufus has been educated in a certain way but also grew up in isolation & is still isolated. Children or people in general start to use that kind of words when they socialize with people or groups that already use those words or type of language. It's called mirroring. If someone like say Sephiroth or Genesis don't use those words, I expect it's the same with Rufus. We also never see him use the f- words canonically. His speech pattern tend be really analytical & sometime sarcastic, even in his mind in the novels.

Tseng won't openly drop it too bcos he's too professional. He might do it but it's also in his mind or when nobody is around.

Rufus' mother: My thought on her depends if she was Shinra sr's legal wife or actually another mistress of his. There's that strange thing going on where the texts only address her as Rufus' mother but never has his Shinra's sr wife. Her presence in the story of the family is barely a thing. There's no photo of her, even in the Shinra museum despite the fact she would have likely been a beautiful wife to expose as an accomplishment for the President. We don't feel if she had influence of Rufus too beside him inheriting some of her physical traits.
My guess for the first time option is that she was a beautiful woman but also likely dangerous or cold-hearted enough to marry a guy like him without batting an eye to all the abuse going on with Shinra. However the marriage grew quickly cold because Shinra sr like having subservient people under him and she was not that. As he also couldn't find the magic he had when he was openly dating Lazard's mother (that he ended up abandonning for political reasons) He also cheated on her. Rufus' mother would have mistresses fired or sometime force them to have abortions when some ended up pregnant & were seemingly interested to gain more status with the unborn child when they were highly in favor. Anything to secure her position and Rufus until she passed away when Rufus was still a kid.
The mistress option, I don't have much thought about her personnality beside the fact she begged Shinra sr to take care when she passed away from her sickness. Shinra sr who was looking for an heir, picked & legitimized Rufus the day she died (hence why he has this warped idea that the password his father used is the date of her passing. It was the big day that made them family & surely his father cared enough about his mother if he decided to legitimize him)
She used to be a cellist.

Elena's inherited a lot of traits of personnality from her mother. She has a fierce temperament that she ended up repressing for a few years when she married her husband & had to spend her time taking care of her kids while he was away. She also hated the fact that they were supposed to live on the plate but ended up in the slums due to the accident of the sector 6 plate, as Shinra refused to relocate them ( employees get to know if they live on the plate or not if they win the house lottery) Her relationship with her husband was not warm either & she felt at time treated like a glorified babysitter.
Her limit was reached when Elena was only 4-5 y old. She loves her daughters but ended up divorcing & leaving their life as she wanted her independance back. That event shaped a lot the girl's vision of loyalty as a results.

React like she often does: showcase anger & pride,scoff & maybe find a way to get revenge by undercuting or insulting the people saying that depending of how bad it is.

If the two of them are already close, it will be one of those cases where she knows she can affirm her beliefs. She doesn't put Rufus on pedestal like many people do. She sees him as a flawed man who also has his own qualities. But he's not the type of person who can build a relationship with just anyone. And turns out that she's one of the rare person out there who match his freak on various domains.

I still sees her seeking for re-assurance & support from him when nobody is looking tho. I think it would be interesting to observe their growth as they learn on that side, as these two comes from places where they didn't felt necessarily supported as they become each other's "safe place"

I don't know really. There's the fact that her motive to personally chase him is gone & we kinda already got the punch/roll down the cliff as a her boss battle mechanic. And Cloud is now completely unhinged too. Him letting her punch him wouldn't make sense knowing Sephiroth's influence is so strong that he almost offed her in the temple.
My hopes tho is that they'll change this & make her the Turks (and maybe Rude & Reno as well) who accompany Rufus in the Northern Cave.
They set her up as someone curious, someone who was actually excited at the idea of finding the promised land, who might ultimately wants answers about what's going on behind-the scenes but also the fact she's worried about Rufus stunts. There's also the fact she's the last Turks you see in Rebirth before you switch to Rufus.
Her deciding to "redeem" herself by personnally following Rufus to the Northern Cave might work better in the context of part 3.
And tbh, I don't care much if it gets cut (it's not important to her & just undercut her character) & if it is replaced by something that doesn't make more disservice to her character but enhanced it instead.

Rufus is not the most physically durable of the group but he knows how to compensate with his smarts & quick reflexes/reaction time. He could win a fight like this. Which could be an interesting (needed) challenge for Elena.
She's smart but her overeagerness causes her to do bad mistakes in combat ( the Elena knuckle is almost worthless due to the time she takes to charge the attack & she misses to the point of falling down cliffes, she sometime need to take a breath because she doesn't spend her energy correctly etc)
So fighting someone like Rufus might push her to think a bit more beforec acting to win, which could be a valuable learning experience for the both of them.