Basically anything can happen: fix $M/F$ of degree $m$ and $f(x) = x^n + f_{n-1} x^{n-1} + \ldots + f_0 \in M[x]$; if $f$ is irreducible (which it will be for "random" $f$), then let $\alpha$ be a root and $K=F(\alpha)$ and then $\operatorname{Norm}_{K/M}(\alpha) = \pm f_0$ and $\operatorname{Tr}_{K/M}(\alpha) = \pm f_{n-1}$. You can choose $f_0$ and $f_{n-1}$ freely (in particular they can generate whatever subfield of $M$ you like) and so the same is true for norm and trace.

Basically anything can happen: fix $M/F$ of degree $m$ and $f(x) = x^n + f_{n-1} x^{n-1} + \ldots + f_0 \in M[x]$; if $f$ is irreducible (which it will be for "random" $f$), then let $\alpha$ be a root and $K=M(\alpha)$ and then $\operatorname{Norm}_{K/M}(\alpha) = \pm f_0$ and $\operatorname{Tr}_{K/M}(\alpha) = \pm f_{n-1}$. You can choose $f_0$ and $f_{n-1}$ freely (in particular they can generate whatever subfield of $M$ you like) and so the same is true for norm and trace.

Basically anything can happen: fix $M/K$ of degree $m$ and $f(x) = x^n + f_{n-1} x^{n-1} + \ldots + f_0 \in M[x]$; if $f$ is irreducible (which it will be for "random" $f$), then let $\alpha$ be a root and $K=F(\alpha)$ and then $\operatorname{Norm}_{K/M}(\alpha) = \pm f_0$ and $\operatorname{Tr}_{K/M}(\alpha) = \pm f_{n-1}$. You can choose $f_0$ and $f_{n-1}$ freely (in particular they can generate whatever subfield of $M$ you like) and so the same is true for norm and trace.

Basically anything can happen: fix $M/F$ of degree $m$ and $f(x) = x^n + f_{n-1} x^{n-1} + \ldots + f_0 \in M[x]$; if $f$ is irreducible (which it will be for "random" $f$), then let $\alpha$ be a root and $K=F(\alpha)$ and then $\operatorname{Norm}_{K/M}(\alpha) = \pm f_0$ and $\operatorname{Tr}_{K/M}(\alpha) = \pm f_{n-1}$. You can choose $f_0$ and $f_{n-1}$ freely (in particular they can generate whatever subfield of $M$ you like) and so the same is true for norm and trace.