Today I called my mother to askcatch up on her news, then I told her that I went to the dentist and that everything is finegoing well with me. Today I called my mother to catch up on her news, then I told her that I went to the dentist and that everything is going well with me.

It's a little unnatural to say "ask her news", though it would work informally!

Today, I called my mother to ask her newsif she had any news to share, then I told her that I went to the dentist and everything is fine with mI'm fine. Today, I called my mother to ask if she had any news to share, then I told her that I went to the dentist and I'm fine.

Today I called my mother to ask her news, thenow she was doing. I told her that I went to the dentist and everything is fine with me. Today I called my mother to ask how she was doing. I told her that I went to the dentist and everything is fine with me.

You almost wrote this correctly. In the US, we don't ask for people's news. Instead, we call to ask how people are doing. The sentence flow is better if this thought is written in two separate sentences.