Does this sentence sound natural and correct to you?

yes is does

No, it sounds like you and the person you are talking to are somewhere other than America. You are requesting him/her to send updates if their name is mentioned in the American news.

If you want an update about their activity in America, you would say 'Please update me of your activity in America.'

If you want to know if they are waiting to find out about going to America, you would say 'Please update me when you get the news about America.'

Sounds right. It could also be "Please update me on any news of you in America." But I think it's really good already.

Yes, it is a good English sentence.