How did Louise know the General's private number?
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How did she know what the General's wife said?
This is a bit complicated to explain, and opinions vary on some of the specific details, but it's crucial to the film's ending so I'll try to answer it simply and briefly:
The Heptapods' "gift" to humanity is their language, By learning it, the translator's brain is essentally "rewired" (a concept loosely based on the Sapir-Whorf hypothesis mentioned in the film) so that they perceive time
non-linearly
, just like the Heptapods do. The upshot of this is that by learning Heptapod, Louise starts experiencing time non-linearly: in effect, she has "memories" of the future. This is how she is able to "remember" her daughter Hannah before she was born, and how she was able to "remember" meeting General Shang before phoning him. Louise therefore knows Shang's number, and what his wife's last words were because she "remembers" him telling her these things in the future.
As an aside: there has been a lot of discussion about a potential paradox here, but without going into it in detail my understanding is that the key point is that from Louise's perspective, she effectively experiences phoning General Shang and meeting him at the party
at the same time
. This would make my references to "remembering" above a bit misleading, but I'm just using that term for convenience.
What did she say to him in Mandarin over the phone?
I'm not sure about the whole conversation (some other posters have tried to translate it, but as others here have observed Adams' Mandarin is apparantly not that great), however the screenwriter Eric Heisserer has confirmed that the last words of Shang's wife translate as: "In war there are no winners, only widows." It was Denis Villeneuve's decision not to provide subtitles for that scene.
https://www.thrillist.com/entertainment/nation/arrival-chinese-line-ending
Did the General change his mind just because he was impressed with what Louise knows about his wife?
Shang himself implies that it was because - as he says - Louise did something that no one else had ever done: she surprised him. I think we're meant to assume that that - combined with the anti-military tone of his wife's comments - was enough to make him halt and take stock.
Did the General show Louise the received calls record in his phone. How could she recognize it was her number, she was using a sat phone?
Not 100% certain from memory, but I think he just showed her his private mobile number. It would not be necessary for her to recognize her call/number, as Shang told her she called him anyway.
IIRC it's already mentioned in the Trivia section, buit at some point I really need to get round to adding the wife's last words to the FAQ - that's if some other kind soul doesn't beat me to it first
"So I've got bullets, but no gun. That's quite Zen."

I am a Chinese, but sorry to say that Amy Adams speaks bad Mandarin, I have no idea what she said as well.
But I read in many post that they say she is saying War does not make winners, only widows. I don't know how they catch that phrase though, maybe I should give this movie a second go.
By the way, I think this is a brilliant movie.

When Louise saw the future, the Chinese General showed her his phone number and whispered to her what his wife said last.
As for what was said, I speak six languages but Mandarin isn't one of them. Alas! Hopefully, someone will come along with answers - I, too, am interested in knowing what was said.