almost n-th power in a field

Your first question is interesting but I don't know any answers.

The answer to your second question is well-known. If a field $F$ has a finite extension which is real-closed then it has a finite extension which is algebraically closed (by adjoining $\sqrt{-1}$ to the real-closed field). But the only fields which have finite extensions $K$ which are algebraically closed are either real-closed or algebraically closed already, and the degree of $K/F$ is $2$ or $1$. This is a theorem of Artin and Schreier and can be found in the more comprehensive standard texts, for instance Lang's Algebra.