I have a result that uses ordered fields. However, I am ignorant of the literature surrounding ordered fields. I have read the basic facts about them, such as those contained in the wikipedia page. However, in the introduction, I would like to provide some motivation.

My result has the property that if it holds for some ordered field k, then it holds for any ordered subfield of k. Does that mean it is enough to prove my result for real-closed ordered fields?

I also read the statement that to prove a 1st order logic statement for a real-closed ordered field, then it is enough to prove it for one real-closed ordered field, such as R for instance. Can someone please provide a reference for that?

A mathematician mentioned to me a classical link between iterated quadratic extensions and ordered fields. What is the precise statement please?

I would also like a number of interesting examples of ordered fields. I know for example of an interesting non-archimedean example using rational functions (that I have learned about from the wikipedia page on ordered fields).

Does anyone know of a survey on ordered fields, or a reference containing answers to my questions above?