As your question is about the difference between AmEng and BrEng, the first thing I should point out is that "go get" is typically American. British English speakers usually say "go and get".
"Getting fresh air", or "getting some fresh air" are pretty universal to both AmEng and BrEng. "Fresh air" means the air outside, as opposed to the air inside a building, which can become stale. Your interpretation "to go get relaxed in the wind" is too literal of a translation. Fresh air does not mean wind - the air can be still, and be fresh; also most people do not find the wind to be relaxing. Getting fresh air can include relaxation, but it can also be for health benefits or be part of some other outdoor exercise.
I would say that the simplest, most idiomatic way to express that you were stepping outside to get some fresh air would be:
I'm going out to get some fresh air.
Informal expressions might include:
- Let's go get some fresh air. (AmEng)
- Let's go and get some fresh air. (BrEng)
- I'm going to go get some fresh air. (AmEng)
- I'm going to go and get some fresh air. BrEng)
These are all idiomatic, although some may criticise the latter two for the use of "going to go".
Looking at your examples, most do not sound right for various reasons. "Take in some fresh air" is idiomatic, but formal.