(1) Both "that" and "who" are totally correct, and I honestly don't see a big difference between them.
(2) You wrote, "the restaurant is only famous between the neighborhood" but the word "between" is strange here. I would use "within" instead, because "between" is usually for when something is in the middle of two things. You should write, "the restaurant is only famous within the neighborhood" or perhaps "the restaurant is only well-known within the neighborhood."
(3) As others have said, "much" is not correct here. But including "them" isn't exactly necessary. Both ways are grammatically correct (with/without them) but have slightly different meanings. See below:
"here's a thread to show them how wrong they are" = You're showing "them". They're seeing it. They feel stupid. ;)
"here's a thread to show how wrong they are" = You're showing the world. The world sees how stupid they are. "They" may or may not see it, it doesn't matter, but you've told the world.