It's okey to use "cám ơn" that way, it's not wrong. However, that's not the way vietnamese people normally answer in this situation. We just say "tôi khỏe"

"Cám ơn" can be said before or after "Tôi khỏe". There is no difference. :D

I think we have a little bit diference. In culture Americe, when someone begin a talk, he says "hey guys/ how are u ?" and the other one say " i'm fine, thanks" that mean he don't want to keep talking. And in Vietnam, if "Cảm ơn" before a sentence that's like same mean American culture. For example , when u walk on the treet, a sales person suddently begin talking with u to sreve his work, u can say " cảm ơn, tôi không muốn" ( i don't want it, thanks)

"Cảm ơn", it is a way we say when we get a question that relate to us. It has two means. Firstly, we want to thank you for focus us. Secondly, it makes the conversation polite and friendly. In English, you say "How are you today?". I can answer that "Yes", "Thanks, I'm so so", "I'm fine now". So you can see the sentence "Thanks, I'm so so". It's the same Vietnamese in this situation.

With Vietnamese culture, you should respond the question first, then you can say "cám ơn". In every situation, if people ask you to care for you, I think saying "cảm ơn" (thanks or thank you) is the best to show the respect for each others.