Learn English – “How much is it important” or just “How is it important” (without much)

If their mother noticed that they have been talking for a long time then it is preferable to use "that much."

But if she just noticed that they are talking then I think she will use "so much."

And in your sentence she said "Don't" which is an order so using "so" is better.

If she used "shouldn't" then "that much" would be better. I think that both are correct, but "so" is more accurate.

If something is very important to somebody, it means that that person feels themselves that it is important, contrary (or fittingly) to what other people thing.

Schoolwork is very important to Bob, as he wants to become a doctor.

Saying something is important for somebody or something means that the thing is important in order for something to be completed, so

Schoolwork is very important for Bob to become a doctor.

Saying Something is important for somebody to do something makes perfect sense, "Doing homework is important for Bob to become a doctor"

You used the word that correctly, although it isn't always required, It makes sense to say "It's important to me for you to be here", or you could say "It's important to be that you are here"