Trouble piecing together LS coupling

[AronYstad]

TL;DR

I've heard a few different statements about LS coupling, but I struggle to see how they relate to each other.

I recently finished my first course in atomic physics, but I don't really feel like I got a good understanding of what LS coupling actually is. At first, we talked about constants of motion and how LS coupling is where we assume that L and S have good quantum numbers, or something similar to that. Then, we talked about addition of angular momenta, and said that with LS coupling, you can add the total L and total S to form J. Later in the course, we talked about heavier atoms and said that when LS coupling doesn't hold anymore, you can for example get transitions with a change in S.

The problem is that I can't find the connections between these statements. So my main questions are:

1. How do the quantum numbers being good relate to L + S = J?
2. What is the mechanism behind why LS coupling stops being a good approximation in larger atoms with stronger interactions?
3. Why can S change in transitions in larger atoms? And why is it very unlikely even when it does happen?

And if you have any more details that would help me understand the concept, please share. I've sent a few messages in the class group chat asking about this, but the others seem to also have the same questions.