"Her children do her credit" means that the way her children behave (or some aspect of the children) reflects well on her – that is, she deserves to be praised or thought well of because it's obvious she did a good job of raising them.

The word credit can be used to give praise, approval or honour. Specifically, it can be used by someone who has been or is responsible for you to receive praise by your good behaviour or successful actions: So in your example, the mothers children could have been really well behaved, kind, generous. If someone observes this they could say that those children do their mother credit. Implying that the mother has done an excellent job of raising them. Hope this helps!

The implication is that she has raised her children well and that she should be proud of them. It's a subtle distinction. There's a related sentence along the same lines where someone might say that someone's children 'are a credit' to their parents.

This sentence is perfectly correct. It means that other people think her children are well behaved, successful at what they do etc. And the reason can be traced back to their mother. She can indeed be proud of them.

Yes, this is idiomatic English. The phrase means that her children are doing things (or living lives) that reflect well upon their mother. For example, if all my children were successful people, my friends might say, "His children do me credit." They also might say, "His children do him proud." Hope this helps!